SOME METRIC ASPECTS IN ALGEBRAIC

SOME METRIC ASPECTS IN ALGEBRAIC GEOMETRY, ON THE AVERAGE

Facultad de Ciencias Universidad de Cantabria

Programa Oficial de Postgrado de Ciencias, Tecnolog´ıa y Computación Máster en Matemáticas y Computación Trabajo Fin de Máster Autor: Mario Fern´ andez Pend´ as

5 de Septiembre de 2012

Director: Luis Miguel Pardo Vasallo

Investigación parcialmente subvencionada por el proyecto MTM2010-16051

Contents Chapter 1. Introduction and Statement of the Main Results 1.1. Introduction 1.1.1. Structure of the Manuscript 1.2. Some basic notions and notations 1.3. Statement of the Main Results 1.3.1. An Arithmetic Poisson Formula 1.3.2. Expected Mahler’s measure of Chow Forms and Elimination Polynomials 1.3.3. Some “harmony” in the Expectation of Mahler’s measure 1.3.4. On expected distance and expected separation

5 5 7 7 10 10 12 13 14

Chapter 2. Basic Notions 2.1. Basic Notations: Bombieri-Weyl metric 2.1.1. Bombieri-Weyl Metric 2.2. Solution Variety and Some Basic Integration Formulae 2.2.1. The Geometry 2.2.1.1. The Solution Variety 2.2.1.2. The over-determined case (m) 2.2.2. A few words about Probability Distributions on H(d) 2.2.3. Normal Jacobians and the Co-area Formula 2.2.4. Double fibration and some integral formulae

19 19 19 20 20 21 21 22 24 25

Chapter 3. Proof of the Main outcomes about expected distances and separations 3.1. Proofs of the statements about expectations, distances and separation bounds 3.1.1. Proof of Theorem 1.3.5 3.1.2. Proof of Theorem 1.3.6 3.1.3. Proof of Theorem 1.3.8 3.1.4. Proof of Theorem 1.3.9

27 27 27 29 30 31

Chapter 4. Arithmetic Poisson Formula and Applications 4.1. Mahler’s measure of multivariate polynomials 4.1.1. Mahler’s measure 4.1.2. Proof of Theorem 4.1.1 4.2. An Arithmetic Poisson Formula 4.2.1. Unitarily invariant height: An Arithmetic Poisson Formula 4.2.2. Proof of Theorem 4.2.1 4.3. On the average Mahler’s measure of the Chow forms and affine elimination polynomials for zero-dimensional varieties

33 33 33 39 41 42 43

Chapter 5. Resumen en Castellano de los contenidos del Trabajo Fin de Máster 5.1. Introducci´ on 5.2. Enunciado de los principales resultados 5.2.1. Algunas nociones b´ asicas y notaciones 5.2.2. Una F´ ormula Aritmética de Poisson 5.2.3. Esperanza de la medida de Mahler de las Formas de Chow y de los Polinomios de Eliminaci´ on 5.2.4. Cierta “armon´ıa” en la esperanza de la medida de Mahler 5.2.5. Esperanza de la distancia y de la separación

51 51 53 53 56

3

47

58 59 60

4

Bibliography

CONTENTS

65

CHAPTER 1

Introduction and Statement of the Main Results 1.1. Introduction This manuscript contains some highly technical results of ”interdisciplinary mathematics”. The main motivation of this research is the Design and Analysis of Efficient Numerical Algorithms in Algebraic Geometry. Thus, the motivation comes at least from three usually distant mathematical frameworks: Computational Complexity, Numerical Analysis and Algebraic Geometry. Most of these results focus on questions related to Smale’s Seventeenth Problem. This problem was stated in the famous list of 18 Problems for the Next Century written by S. Smale, after a proposal of V.I. Arnold (cf. [Sm, 00]). Problem 1.1.1 (Smale’s 17th Problem). Can a zero of n complex polynomial equations in n unknowns be found approximately, on the average, in polynomial time with a uniform algorithm? This problem was solved in [BP, 09a] (cf. also [BP, 11a]) by exhibiting a Las Vegas ramdomized algorithm that solves multivariate complex polynomial equations in average quadratic time in the input length (O(N 2 )). There are still beautiful open questions which, on the one hand, aim to improve technically the proposed solution and, on the other hand, wish to pursuit the investigations motivated by this solution. This manuscript aims to work in both directions. One of the main features of the known solutions to Smale’s 17th Problem was already on the basis of the work developed in colaboration of M. Shub and S. Smale between 1981 and 1995 (which culminated in the famous series of manuscripts known as “Bézout series”, which go from [SS, 93] to [SS, 94]). These authors were not able to provide an efficient algorithm to solve multivariate polynomial equations (and this motivated Smale’s statement of his 17th Problem). However, they pointed some of the hints to find such an algorithm. Among them, we may recall the following ones: (1) Use the non-linear condition number µnorm to bound the complexity of algorithms based on homotopic deformation. (2) Use techniques from Integral Geometry to deal with average properties of quantities related to the complexity. In fact, the solution of Smale’s 17th Problem in [BP, 09a] and [BP, 11a] was partially based on introducing new techniques in Integral Geometry, when related with condition numbers and algebraic varieties. For instance, a key ingredient was the study of the average value of µ2norm along great circles of a complex sphere given by Bombieri-Weyl’s Hermitian product. This manuscript also follows this research framework by studying expectations of other quantities related to algebraic varieties. For instance, we study expectations of some metric properties of algebraic varieties. In fact in Subsection 1.3.4 we exhibit the average distance between two randomly chosen complete intersection projective varieties (cf. Subsections 3.1.1, 3.1.2 for the proofs of these statements). The main motivation of this question is the Nullstellensatz-like Numerical Problem stated as Problem 1.3.2 below. Note that, for instance, Theorem 1.3.5 shows that the expected average distance between two randomly chosen complete intersection projective algebraic varieties coincides with the observable diameter of Pn (C) by M. Gromov as in [Gr, 99] and references therein. We also study the expected separation of the zeros of a randomly chosen zero-dimensional complete intersection variety. This is a classical issue in numerical-symbolic treatment of polynomial equations and their 5

6

1. INTRODUCTION AND STATEMENT OF THE MAIN RESULTS

solutions, and it is also motivated by the presence of redundancies in the output of a nonuniversal solver (cf. the statement of Problem 1.3.3 and [BP, 06] or [CaGiHeMaPa, 03] for non-universal polynomial equation solvers). Our main outcomes on the separation bounds are stated in Subsection 1.3.4 and the proofs of this statements are given in Subsections 3.1.3 and 3.1.4. Our main outcome on the average separation bounds is Theorem 1.3.8 which shows that the average expected separation of a randomly chosen complete intersection zerodimensional variety is polynomial in the input length, whereas the usual worst case known bounds are doubly exponential in the number of variables. As a reader may easily conclude, the cited new outcomes necessarily involve aspects from Algebraic Geometry, Numerical Analysis, Integral Geometry, Measure Theory, Probability, Intersection Theory and others thus strenghting the interdisciplinary nature of this manuscript. In spite of the beauty and potential applicability of our results on average distance and separation, they are not the main outcomes of the manuscript. The main outcome of these pages is the Arithmetic Poisson Formula, stated as Main Theorem 1.3.1 below (see Subsection 4.2.2). This central statement of this manuscript adds two new research frameworks to the interdisciplinary nature of these pages: Diophantine Geometry and Elimination Theory. From Diophantine Geometry we consider the notion of height of polynomials, projective points with coordinates in number fields and diophantine algebraic varieties. This notion was initially developed by A. Weil to measure the size of a projective point x ∈ Pn (k), where k is a number field. Then, inspired by the Bézout Inequality, several authors used the notion of height of a diophantine variety as the key notion to prove an Arithmetic Bézout Inequality (cf. [Ph, 86], [Ph, 91], [Ph, 95], [BGS, 94], for instance). Here we study the height of the (n+1) multi-variate resultant variety R(d) (cf. Corollary 1.3.2). (n+1)

The multi-variate resultant R(d) is the central subject of Elimination Theory. It is a projective algebraic hyper-surface of the space of coefficients of an over-determined system of homogeneous polynomials (f0 , . . . , fn ) of respective degrees determined by the list (d) = (n+1) (d0 , . . . , dn ). This multi-variate resultant variety R(d) is the set of common zeros of an (n+1)

irreducible multi-homogeneous polynomial Res(d)

with integer coefficients and, hence, a

(n+1) Res(d)

diophantine object too. The polynomial is known as the multi-variate resultant and its main feature is given by the following property: Given f0 , . . . , fn ∈ C[x0 , . . . , xn ], n + 1 homogeneous polynomials. Let (d) = (d0 , . . . , dn ) be the list of degrees. Then, the over-determined system of equations f = (f0 , . . . , fn ) has a (n+1) common projective zero if and only if Res(d) (f0 , . . . , fn ) = 0. Namely, (n+1)

∃ζ ∈ Pn (C), f0 (ζ) = · · · = fn (ζ) = 0 ⇔ Res(d)

(f0 , . . . , fn ) = 0.

Note that the left hand of this equivalence is a first order formula which contains, as particular instances, all known NP-complete problems. It is just a Projective Nullstellensatz problem. (n+1) In other words, the knowledge and computation of Res(d) is a key ingredient to understand many intractable problems of Computational Complexity. Here we are modest since we just want to find sharp bounds on the logarithmic height of the multivariate resultant. To this end, we may firstly apply the Arithmetic Bézout Inequality. However, this attempt does not yield sharp upper bounds (cf. Section 4.2.2 below and Equation (1.3.1)). We thus proceed by a different approach. Multivariate resultants satisfy (n+1) a famous classical formula known as Poisson Formula. This formula relates Res(d) with (n)

Res(d0 ) , where (d) = (d0 , . . . , dn ), (d0 ) = (d1 , . . . , dn ). This relation is, for instance, helpful

1.2. SOME BASIC NOTIONS AND NOTATIONS (n+1)

to compute the degree of Res(d)

which is known to be equal to

n Y X

7

dj . The reader may

i=0 j6=i

follow a precise statement of Poisson Formula in [CD, 05] and references therein. Combining Poisson Formula and standard notions of height does not yield an Arithmetic Poisson Formula (as it happens with degree bounds). The reason is that the usual notion of height (as the one by Philippon in [Ph, 91]) just leads to inequalities and, hence, it does (n+1) (n) not help to have an equality which expresses the relation between h(Res(d) ) and h(Res(d0 ) ). We succeeded to produce an Arithmetic Poisson Formula by replacing the usual notion of height by a variation based on Bombieri-Weyl’s Hermitian product. We call this new height (n+1) the unitarily invariant height hU (Res(d) ) and it is much better suited to understand the relations existing between the height of the zeros and the height of the equations. The Arithmetic Poisson Formula is then stated as Main Theorem 1.3.1 below and the proof is described in Section 4.2. Precise notions and notations required to understand Main Theorem 1.3.1 are described in Subsection 1.3.1. The notion of unitarily invariant height is introduced in Subsection 1.3.1 and discussed in more detail in Subsection 4.2.1. From our bounds on the unitarily invariant height of the multi-variate resultant we (almost) inmediately conclude sharps bounds on the Mahler’s measure of two central objects of Elimination Theory: Chow polynomials (also known as van der Waerden’s U-resultant), and Elimination polynomials. These sharp bounds are stated as Corollary 1.3.3 and these two kinds of polynomials are discussed in Subsection 1.3.2. The proofs are exhibited in Section 4.3. In the path to prove the Arithmetic Poisson Formula we also prove a technical result which due to the beauty of its statement we have called “harmony” on the expected Mahler’s measure. This is the Main Theorem 1.3.4 below, whose proof is exhibited in Section 4.1. This statement simply shows an equality between the expected Mahler’s measure of a random polynomial and a difference of two harmonic numbers. This result gives us hopes to get a sharp estimate (or a precise equality) of the expected Akazuka zeta function of a random polynomial, but this is yet conjectural. 1.1.1. Structure of the Manuscript. The manuscript is structured as follows. Section 1.3 is devoted to introduce some basic notations and notions which suffice to read the main outcomes of these pages which, by the way, are also stated in this Section. This section is also divided into subsections each one devoted to state the main aspects discussed in this Introduction. Chapter 2 is devoted to establish some basic facts and some technical tools used along the proofs. Section 2.1 introduces Bombieri-Weyl’s Hermitian product, whereas Section 2.2 of this chapter introduces the main geometric tools: the Solution Variety, the Co-area Formula and some aspects on the probability distributions used along these pages. Chapter 3 is devoted to prove the statements concerning average distance and expectations (which are stated in Subsection 1.3.4). Chapter 4 is devoted to prove the Arithmetic Poisson Formula and some of its consquences. So, Section 4.1 recalls Mahler’s measure of multivariate polynomials and proves Main Theorem 1.3.4 on the ”harmony” of the expected Mahler’s measure. Section 4.2 introduces the unitarily invariant height of the multi-variate resultant and proves Main Theorem 1.3.1 (the Arithmetic Poisson Formula). Section 4.3 is devoted to prove the sharp estimates on the expected Mahler’s measure of Chow forms and Elimination polynomials in the zero-dimensional complete intersection case. 1.2. Some basic notions and notations (n+1)

Let n, m ∈ N be two positive integers. For every degree bound d, we denote by Hd the complex vector space of all homogeneous polynomials in C[X0 , . . . , Xn ] of degree d. With (n+1) the same notations, Pd is the complex vector space of all polynomials of degree at most d in C[X1 , . . . , Xn ]. Both vector spaces are obviously isomorphic of dimension d+n and n

8

1. INTRODUCTION AND STATEMENT OF THE MAIN RESULTS (n+1)

(n+1)

the isomorphism is given by the mapping a : Hd −→ Pd , which associates to every (n+1) (n+1) a homogeneous polynomial f ∈ Hd its “affinization” f := f (1, X1 , . . . , Xn ) ∈ Pd . We (n+1) simply write Hd and Pd (omitting the super-script ) when no confusion about the number of variables arises. Let us fix the number n + 1 of variables along this section. For every list of positive degrees Q (m) (d) = (d1 , . . . , dm ) ∈ Nm , we denote by H(d) := m i=1 Hdi the complex vector space of all lists of m polynomials f = (f1 , . . . , fm ), where fi ∈ C[X0 , . . . , Xn ] is an homogeneous polynomial Q (m) of degree di for every i, 1 ≤ i ≤ m. We analogously denote by P(d) := m i=1 Pdi the set of lists of m affine polynomials g = (g1 , . . . , gm ), where gi ∈ C[X0 , . . . , Xn ] is an affine polynomial (m) of degree at most di . The affinization obviously defines and isomorphism between P(d) and (m)

H(d) . Namely, the following mapping is an isomorphism of complex vector spaces: (m)

(m)

H(d) −→ P(d) . a (f ) = (f1 , . . . , fm ) 7−→ ( f ) := (a f1 , . . . ,a fm ) The complex dimension of both vector spaces satisfies: (m)

(m)

dimC (H(d) ) = dimC (P(d) ) :=

m X di + n . n i=1

We sometimes consider the complex projective space defined by any of these spaces. We (m) denote by P(H(d) ) this complex vector space and we denote by N(d) the complex dimension of this complex projective space. When no confusion arises we simply denote by N this dimension. (m) For every list f = (f1 , . . . , fm ) ∈ H(d) , let VP (f ) ⊆ Pn (C) be the complex projective variety of their common zeros: VP (f ) = {X ∈ Pn (C) : fi (X) = 0, 1 ≤ i ≤ m} ⊆ Pn (C), (m)

Similarly, for every list g = (g1 , . . . , gm ) ∈ P(d) , we may consider the affine algebraic variety VA (g) ⊆ Cn of its common zeros: VA (g) = {X ∈ Cn : gi (X) = 0, 1 ≤ i ≤ m} ⊆ Cn . Let ϕ0 be the standard embedding of Cn into Pn (C), (1.2.1)

ϕ0 :

Cn −→ Pn (C) \ {X0 = 0}. (X1 , . . . , Xn ) 7−→ (1 : X1 : . . . : Xn )

Observe that VA (a f ) can be identified with VP (f ) ∩ (Pn (C) r {X0 = 0}). Then, we may write (m) VA (a f ) = ϕ−1 0 (VP (f )) for every f ∈ H(d) . In what follows, we denote Q by d := max{di : 1 ≤ i ≤ m} the maximum of the degrees in the list (d), and by D(d) := m ezout number associated with the degree list i=1 di we denote the B´ (d). (m) As in [SS, 93] or [BCSS, 98] (Sec. 12.1) we may equip H(d) with the unitarily invariant Bombieri-Weyl’s Hermitian product, (m)

(m)

h·, ·i∆ : H(d) × H(d) −→ C. p This Hermitian product is defined in Subsection 2.1.1 below. We denote by k·k∆ := h·, ·i∆ (m) the norm induced by this Hermitian product and we denote by S(H(d) ) the sphere of radius (m)

one centered at the origin in H(d) , defined by this Hermitian product. Namely, (m)

(m)

S(H(d) ) := {f ∈ H(d) : kf k2∆ = 1}.

1.2. SOME BASIC NOTIONS AND NOTATIONS

9

In what follows we shall use the harmonic series and harmonic numbers to express our statements. Thus, let us recall here some basic facts about harmonic numbers. We denote by Hr the r-th harmonic number. Namely, Hr :=

r X 1 . k k=1

Recall that the n-th Harmonic number Hn satisfies: lim (Hn − log(n)) = γ,

n7→∞

where γ is the Euler-Mascheroni number. Recall also that (Hn − log(n)) = γ +

∞ X ζ(m, n + 1) , m

m=2

where ζ(m, n + 1) is Hurwitz zeta function. Additionally, we also have (Hn − log(n)) = γ +

1 1 1 + − ε, − 2 2n 12n 120n4

for 0 0. A second feature of this Arithmetic Poisson Formula is that it yields upper bounds for the probability (in Gaussian distribution) that the multi-variate resultant grows too fast. This is stated in Corollary 4.2.3 below. 1.3.2. Expected Mahler’s measure of Chow Forms and Elimination Polynomials. A third application of this Arithmetic Poisson Inequality yields sharp upper bounds on the expected Mahler’s measure of Chow forms and Elimination polynomials with respect to complete intersection zero-dimensional algebraic varieties (cf. Section 4.3). We now return to some elementary Elimination aspects in the zero-dimensional complete (n) intersection case. As above, let H(d) be the class of all systems of homogeneous polynomial (n)

equations of respective degrees given by the degree list (d) = (d1 , . . . , dn ). Let γ∆ be (n) the Gaussian distribution in H(d) induced by Bombieri-Weyl’s Hermitian product. There ˜ ⊆ H(n) of probability zero (with respect to γ (n) and also with respect to is a subvariety Σ (d) (n) H(d) )

∆

(n) H(d)

˜ the varieties VP (f ) ⊆ Pn (C) and Lebesgue measure in such that for all f ∈ r Σ, a n VA ( f ) ⊆ C are smooth zero-dimensional varieties (i.e. a finite set of smooth points).

1.3. STATEMENT OF THE MAIN RESULTS

13

(n) ˜ the cardinal satisfies: Moreover, for every f ∈ H(d) r Σ

](VP (f )) = ](VA (a f )) = D(d) =

n Y

di .

i=1

˜ is the cone over the discriminant variety Σ ⊆ P(H(n) ). We simply say “f a The variety Σ (d) (n) (n) ˜ generic system in H ” to mean f ∈ H r Σ, and we may claim that the varieties VP (f ), (d)

(d)

VA (f ) are (projective and affine, respectively) zero-dimensional complete intersection varieties (n) for a generically given system f ∈ H(d) . (n)

Two are the main elimination polynomials for a generically given f ∈ H(d) : • The Chow form (also van der Waerden’s U-resultant): Let us introduce some new variables {U0 , . . . , Un } and the linear form U = U0 X0 + . . . + Un Xn . The Chow form (also U-resultant) of VP (f ) is the homogeneous polynomial: (n+1)

ChowVP (f ) := Res(d)

(f1 , . . . , fn , U) ∈ C[U0 , . . . , Un ].

(n) ˜ be • Elimination polynomial: With the same notations as above, let f ∈ H(d) r Σ given and let p ∈ C[X1 , . . . , Xn ] be an additional affine polynomial. We define the elimination polynomial of p with respect to VA (a f ) as the characteristic polynomial χp,VA (a f ) (T ) ∈ C[T ] given by the following identity: Y χp,VA (a f ) (T ) := (T − p(ξ)). ξ∈VA (a f )

We now wish to find estimates for the following two quantities: ECh := Ef ∈γ (n) [mS(H (n+1) ) (ChowVP (f ) )], 1

∆

and Eχp := Ef ∈γ (n) [mS 1 (χp,VA (a f ) )], ∆

(n)

(n)

where γ∆ is the Gaussian distribution in H(d) induced by Bombieri-Weyl’s Hermitian product and norm. Namely, we wish to compute estimates for the logarithmic Mahler’s measure of Chow forms (n) and elimination polynomials for randomly chosen f ∈ H(d) . Moreover, in many cases we can (n+1)

also be interested on the expected value of Eχp for a randomly chosen p ∈ Hd0 to the Gaussian distribution

(1) γ∆

in

(n+1) Hd0 .

with respect

Namely, we are also interested on bounding

Eχ := Ep∈γ (1) [Eχa p ], ∆

where

ap

∈ C[X1 , . . . , Xn ] is the affinization of p as above.

Corollary 1.3.3. With the same notions and notations as above, we can conclude: Qn di ECh ≤ i=1 (log n + c), 2 and Qn Qn ( i=1 di ) d0 Hn 1 i=0 di Eχ ≤ ≤ log n + γ + O . 2 2 n 1.3.3. Some “harmony” in the Expectation of Mahler’s measure. Mahler’s measure has been used as an effective measure of the size of polynomials. It has been considered better suited than the canonical norm k·k2 because of its good behaviour with respect to the product of polynomials (namely M (f · g) = M (f ) · M (g)). Bombieri-Weyl’s norm is the only serious competitor since satisfies Bombieri’s inequality (cf. Proposition 2.1.2 below). Nevertheless, Mahler’s measure remains as the main archimedean measure of polynomials with coefficients in a number field and, hence, Mahler’s measure is the main quantity to study in Arithmetic/Diophantine Geometry.

14


The logarithm of the Mahler’s measure is, in fact,Qan expectation of the logarithm of the polynomial along the product of complex circles ni=1 S 1 ⊆ Cn . This is a feature that Mahler’s measure shares with Bombieri-Weyl’s norm. In the early nineties, P. Philippon (cf. [Ph, 91], [Ph, 95]) introduced some variations of the logarithmic Mahler’s measure, replacing the product of circles by products of spheres and even the sphere S 2n−1 ⊆ Cn . Its main outcome in [Ph, 91] was Theorem 4.1.2 below. This Theorem by Philippon just compares the Q expected values of the logarithmic Mahler’s measures when we choose either the product ni=1 S 1 or the sphere S 2n−1 . In this manuscript we prove the harmonic behaviour of the expected logarithmic Mahler’s (n+1) measure of a randomly chosen polynomial f ∈ Hd . Namely, (n+1)

(n+1)

Main Theorem 1.3.4. Let S(Hd ) be the unit sphere in Hd with respect to Bombierin Weyl’s Hermitian product and let ϕ0 : C → Pn (C) the canonical embedding of the affine (n+1) be the complex dimension of Hd . Let us space into its projective closure. Let R := d+n n define the expectation: E := Ef ∈S(H (n+1) ) [EPn (C) [log |a f ◦ ϕ−1 0 |]]. d

We have: (1.3.2)

d 0≤E= 2

HR 1 Hn − ≤ d(log(n) + γ) + O . d n

1.3.4. On expected distance and expected separation. As a collateral product of our technical studies we also obtain improvements on the study of several aspects relating projective (Fubini-Study) distance and complete intersection algebraic varieties. We assume that Pn (C) is endowed with its Fubini-Study metric structure and dR : Pn (C) × Pn (C) → R+ denotes the Riemannian distance. The Riemannian distance dR : Pn (C)×Pn (C) → R+ is the distance defined by the Riemannian structure of Pn (C). Namely, let π : Cn+1 r {0} → Pn (C) be the canonical projection onto the complex projective space. Given x, y ∈ Pn (C) and x1 , y1 ∈ Cn+1 r {0} such that π(x1 ) = x, π(y1 ) = y, the Riemannian (also called Fubini-Study) distance dR (x, y) is defined by dR (x, y) = arccos

|hx1 , y1 i| , kx1 k2 ky1 k2

where h·, ·, i : Cn+1 × Cn+1 → C is the canonical Hermitian product in Cn+1 , and kx1 k2 , ky1 k2 are the respective norms with respect to this Hermitian product. As in [BCSS, 98] we may also introduce the “projective” distance between two projective points x, y ∈ Pn (C) by the following identity: dP (x, y) := sin dR (x, y). Using this projective distance we may study several different quantities relating distance between zeros and algebraic varieties. In these pages, we just discuss two main classes of questions. The first question is based on the following classical Problem: (m)

(s)

Problem 1.3.1 (Hilbert’s Nullstellensatz). Let f ∈ H(d) and g ∈ H(d0 ) be two systems of homogeneous polynomial equations. Assume that the projective varieties VP (f ) and VP (g) are smooth complete intersections of respective co-dimensions m and s. Then, decide whether VP (f ) ∩ VP (g) is empty or not. Note that the multi-variate resultant was defined to answer questions as this Hilbert’s Nullstellensatz. In fact, assume m = n and s = 1 in the statement of Problem 1.3.1 above. Now, (n) (n+1) given f = (f1 , . . . , fn ) ∈ H(d) and g ∈ Hd0 , Hilbert’s Nullstellensatz has a negative answer if and only if (n+1) Res(d0 ) (g, f1 , . . . , fn ) = 0, where (d0 ) = (d0 , d1 , . . . , dn ) and (d) = (d1 , . . . , dn ). From numerical analysis we have a different approach to this Problem.


15 (m)

(s)

Problem 1.3.2 (Numerical Analysis Nullstellensatz-like). Let f ∈ H(d) and g ∈ H(d) be two systems of homogeneous polynomial equations. Let z ∈ Pn (C) be an approximate zero of f with associated zero ζ ∈ VP (f ). Decide whether ζ ∈ VP (g) or not. Recall that an approximate zero z ∈ Pn (C) of f with associated zero ζ ∈ VP (f ) is a point such that 1 dP (Nfk (z), VP (f )) ≤ dP (Nfk (z), ζ) ≤ 2k−1 , 2 where Nfk (z) ∈ Pn (C) is the k-th iteration of projective Newton’s operator applied to z. Thus, if VP (f ) ∩ VP (g) = ∅, lower bounds for the minimum distance between VP (f ) and VP (g) could be used to control the number of iterations required before we get “far enough” from VP (g) (cf. [BCSS, 98], [BP, 09b], [BP, 11b] for discussions about these ideas and [KP, 96], [HMPS, 00] or [KPS, 01] for other approaches to the Nullstellensatz). We study two main quantities: Z 1 dP (x, y)dνn−m (x)dνn−s (y), δav [f, g] := νn−m [VP (f )]νn−s [VP (g)] VP (f )×VP (g) dP (VP (f ), VP (g)) := min{dP (x, y) : x ∈ VP (f ), y ∈ VP (g)}, where dνn−m and dνn−s are the canonical Haussdorff measures of respective dimensions n−m and n − s (cf. [BP, 07] for more details) and νn−m [VP (f )], νn−s [VP (g)] are respectively the volumes of VP (f ) and VP (g) with respect to these Haussdorff measures. The quantity δav [f, g] is the expected distance of two randomly chosen points x ∈ VP (f ) and y ∈ VP (g). As VP (f ) and VP (g) are both compact sets, dP (VP (f ), VP (g)) is certainly attained. Note that if m + s ≤ n, and the varieties VP (f ) and VP (g) are of co-dimensions m and s respectively, then VP (f ) ∩ VP (g) 6= ∅ and dP (VP (f ), VP (g)) = 0. Thus, the unique interesting cases of study are those where m + s ≥ n + 1. In this case, VP (f ) ∩ VP (g) = ∅ for randomly chosen f and g. We prove the following two statements: Theorem 1.3.5. With these notations, we have: EP(H(m) )×P(H(s) ) [δav (f, g)] = (d)

(m)

(d0 )

1−

1 2n + 1

,

(s)

(m)

where P(H(d) ) × P(H(d0 ) ) is the product of the two canonical distributions in P(H(d) ) and (s)

P(H(d0 ) ) induced by their Riemannian structure (cf. Subsection 2.2.2 below). Note that this equality holds independently of the values of m and s and independently of the fact that VP (f ) ∩ VP (g) is either empty or not. This result is also surprising since the expected distance we obtain coincides with the observable diameter of the complex projective space in Gromov’s terms (cf. [Gr, 99], chapter 3 1/2, and its references). Theorem 1.3.6. With the same notations, assuming m = n and s ≥ 1, the following in(s) equality holds for every fixed g ∈ H(d0 ) such that VP (g) is smooth of co-dimension s and VP (f ) ∩ VP (g) = ∅. 2s − 1 (1.3.3) Ef ∈P(H(n) ) [dP (VP (f ), VP (g))] ≥ 2s , (d) D(d) + 2 deg(VP (g))D(d) es where e is the basisQof the natural logarithm, deg(VP (g)) is the degree of the projective variety VP (g) and D(d) = si=1 di is the Bézout number associated to the list (d) = (d1 , . . . , dn ). In particular, we have 2s − 1 , EP(H(m) )×P(H(s) ) [dP (VP (f ), VP (g))] ≥ Q d0 e2 (d) (d0 ) D(d) 1 + 2 si=1 si 2 where (d0 ) = (d01 , . . . , d0s ) is the list of degrees associated to the second class of equations. As immediate corollary we conclude: Corollary 1.3.7. With the same notations as above, we have:

16


(1) For s = 1, the following inequality holds: E(f,g)∈P(H(n) )×P(H(n+1) ) [dP (VP (f ), VP (g))] ≥ (d0 )

(d)

1 . D(d) (1 + 2e2 d0 )

(2) For s ≥ 3, the following inequality holds: E(f,g)∈P(H(n) )×P(H(s) ) [dP (VP (f ), VP (g))] ≥ (d0 )

(d)

5 . D(d) (1 + 2D(d0 ) )

Another classical subjects in polynomial equation solving is the study of the lower bounds for the separation of the zeros of a system of equations. This is relevant both in the methods based on binary search (divide and conquer methods) and in those based on homotopy deformation techniques of non-universal type (i.e. those methods that do not follow all solutions along the prescribed path as those in Shub-Smale conjecture in [SS, 94], or those based on questor sets as in [BP, 09a] and [BP, 11a]). A question of this class is the following one: (n)

Problem 1.3.3 (Redundancy in Non-universal Numerical Solving). Let f ∈ H(d) be a given system of n multivariate homogeneous polynomial equations. Suppose you are also given two points z1 , z2 ∈ Pn (C) such that both of them are approximate zeros of f with associated zeros ζ1 , ζ2 ∈ VP (f ) respectively. Decide whether ζ1 = ζ2 or not. As both z1 and z2 are approximate zeros of f we obviously have dP (ζ1 , ζ2 ) ≤ dP (Nfk (z1 ), ζ1 ) + dP (Nfk (z2 ), ζ2 ) + dP (Nfk (z1 ), Nfk (z2 )) ≤ ≤

2 22k−1

+ dP (Nfk (z1 ), Nfk (z2 )).

If ζ1 = 6 ζ2 , there is a quantity sep(f ) (the separation of VP (f )) such that sep(f ) ≤ dP (ζ1 , ζ2 ) and, hence, we conclude 2 sep(f ) − 2k−1 ≤ dP (Nfk (z1 ), Nfk (z2 )). 2 In particular, lower bounds for sep(f ) are essential to determine the number k of Newton’s iterations required to decide whether ζ1 6= ζ2 . One of the most popular lower bound on the separation is the famous DMM bound, introduced for the case of univariate systems by J. Davenport (cf. [Da, 88]), who attributes it to ideas of Kurt Mahler and M. Mignotte (cf. [Mg, 95]). These bounds are extendable to the multivariate case in various directions. We suggest the manuscript [EMT, 10] for recent results in these estimates and further references. Among these generalizations we wish to remark [De, 97]. Another example on how to extend Mignotte’s counter-example to the multivariate affine case may be found in [CHMP, 01]. All these statements tend to show a lower bound for the separation which is doubly exponential n in the number of variables (i.e. Ω(22 ) as lower bound). As this lower bound is obtained in the worst case, here we want to explore its average value. We just restrict ourselves to the projective case, although the affine case is also a subject to be treated in forthcoming works. We also restrict ourselves to the zero-dimensional case. (m)

Definition 1.3.4 (Separation bounds). Let f = (f1 , . . . , fn ) ∈ H(d) be a sequence of n homogeneous polynomial equations and assume that the projective variety VP (f ) ⊆ Pn (C) is zero-dimensional (i.e. a finite set of points). (1) We define the average separation among zeros of f by the following equality: X 1 sepav (f ) := dP (ζ, ζ 0 ). ]VP (f )(]VP (f ) − 1) ζ, ζ 0 ∈ VP (f ) ζ 6= ζ 0 (2) We also consider the separation bound as usual: sep(f ) := min{dP (ζ, ζ 0 ) : ζ, ζ 0 ∈ VP (f ), ζ 6= ζ 0 }.


17

We prove the following statements: Theorem 1.3.8. With these notations, the following inequality holds: √ √ s 6(3 − 7) 1 Ef ∈P(H(n) ) [sepav (f )] ≥ , 3 8 d (N + 1/2)n (d) where d = max{di : 1 ≤ i ≤ n}, (d) = (d1 , . . . , dn ) and N = N(d) is the dimension of the (n)

complex projective space P(H(d) ). Note that this lower bound is no more doubly exponential in the number of variables but a polynomial in d, n and the input length N . As for the separation bound, we also prove: Theorem 1.3.9. With these notations, we also have: √ 3− 7 (N + 1/2)−1/2 , Ef ∈P(H(n) ) [sep(f )] ≥ 4eD(d) d3/2 (d) Q Where d, n and N are as in the previous Theorem and D(d) = ni=1 di is the Bézout number. Note that, this time, the lower bound is simply exponential in the number of variables and no more doubly exponential as in the worst case DMM bound.

CHAPTER 2

Basic Notions 2.1. Basic Notations: Bombieri-Weyl metric In this Section we continue with the basic notions and notations to be used along forthcoming pages. 2.1.1. Bombieri-Weyl Metric. As in [SS, 93] or [BCSS, 98, Sec. 12.1], we may (m) equip H(d) with the unitarily-invariant, Bombieri-Weyl’s Hermitian product. Let f, g ∈ (n+1)

Hd be two homogeneous complex polynomials of degree d in n + 1 variables and assume that the following are their respective monomial expansions: X aµ X0µ0 · · · Xnµn , f= µ∈Nn+1 |µ|=d

g=

bµ X0µ0 · · · Xnµn ,

X µ∈Nn+1 |µ|=d

Nn+1

where µ = (µ0 , . . . , µn ) ∈ and |µ| = µ0 + . . . + µn , ∀µ ∈ Nn+1 . We define the Hermitian product hf, gid by the following identity: X d −1 hf, gid := aµ bµ , µ n+1 µ∈N

|µ|=d

where

d d! , = µ0 ! · · · µn ! µ is the multi-nomial coefficient and · denotes complex conjugation. For every polynomial p (m) f ∈ Hd we denote by ||f ||d := hf, f id the Bombieri-Weyl’s norm of f . This norm can be extended to lists of polynomials in the following form. For every degree list (d) := (d1 , . . . , dm ) we extend this Hermitian product in the obvious way. Namely, if (m) (m) f = (f1 , . . . , fm ) ∈ H(d) and g = (g1 , . . . , gm ) ∈ H(d) , then we define hf, gi∆ :=

m X

hfi , gi idi .

i=1 (m)

We denote by ||f ||∆ the norm of some f ∈ H(d) associated to this Hermitian product. We (m)

(m)

will denote by S(H(d) ) the sphere of radius one in H(d) with respect to this metric. Namely, (m)

(m)

S(H(d) ) := {f ∈ H(d) : ||f ||2∆ = 1}. (m)

Similarly, for every degree list (d) := (d1 , . . . , dm ) we denote by S(d) the product of spheres (with Bombieri-Weyl metric) given by the following identity: (m) S(d)

:=

m Y

(n+1)

S(Hdi

),

i=1

where (n+1)

S(Hdi

(n+1)

) := {f ∈ Hdi 19

: ||f ||di = 1}.

20

2. BASIC NOTIONS

The following well–known statement shows that Bombieri-Weyl’s norm is simply an expectation. Proposition 2.1.1 (Bombieri-Weyl’s norm (cf. [De, 06], for instance)). For every homoge(n+1) neous polynomial f ∈ Hd , its Bombieri’s norm satisfies: Z d+n 1 2 |f (z)|2 dνS (z). ||f ||d := νS [S 2n+1 ] S 2n+1 n Moreover, this Proposition yields another proof of the unitary invariance of Bombieri–Weyl’s Hermitian Product. Proposition 2.1.2 ([BCSS, 98], for instance). Let U(n + 1) be the unitary group of (n + (m) 1) × (n + 1) complex matrices. Let us consider the following action of U(n + 1) on H(d) : (m)

(2.1.1)

(m)

U(n + 1) × H(d) (U, (f1 , . . . , fm ))

−→ H(d) 7−→ (f1 ◦ U ∗ , . . . , fm ◦ U ∗ ),

where f ◦ U ∗ denotes composition. Then, this action is isometric with respect to Bombieri(m) Weyl’s Hermitian product. Namely, for every f, g ∈ H(d) and for every U ∈ U(n + 1) the following equality holds: hf, gi∆ = hf ◦ U ∗ , g ◦ U ∗ i∆ . (n+1)

Remark 2.1.1. We may also consider the canonical Hermitian product and norm on Hd (n+1) Let f ∈ Hd be a homogeneous polynomial given by its coefficients: X (n+1) f := . aµ X0µ0 · · · Xnµn ∈ Hd

.

µ=(µ0 ,...,µn )

• The weight (also called the Taxicab norm) is given as wt(f ) := ||f ||1 := R+ . 1/2 P 2 |a | ∈ R+ . • The standard Hermitian norm is given by ||f ||2 := µ µ

P

µ |aµ |

∈

The following inequalities relate Bombieri-Weyl’s norm and the usual ones: d + n 1/2 ||f ||d ≤ wt(f ), ||f ||d ≤ ||f ||2 . n (n+1)

(n+1)

Remark 2.1.2. Let f ∈ Hd be a homogeneous polynomial and let g := a f ∈ Pd “affinization”. Assume g decomposes into homogeneous components as: g :=

d X

be its

gi (X1 , . . . , Xn ),

i=0

where gi is homogeneous of degree i. Then, we have (2.1.2)

||f ||2d

=

d X i=0

||gi ||2i

−1 d . i

2.2. Solution Variety and Some Basic Integration Formulae 2.2.1. The Geometry. Some of the main advances in [SS, 93, SS, 93b, SS, 96, SS, 94] are due to the smart exploration of a geometric structure related to the polynomial system solving: the solution variety.

2.2. SOLUTION VARIETY AND SOME BASIC INTEGRATION FORMULAE

21

2.2.1.1. The Solution Variety. As in [SS, 93], following the notations of [BP, 11a] and also the notations of previous sections, let us fix the set {X0 , . . . , Xn } of homogeneous variables and let m ∈ N be a positive integer. Let (d) = (d1 , . . . , dm ) be a list of positive degrees. (m) (m) We define the projective solution variety V(d) ⊆ P(H(d) ) × Pn (C) by the following equality: (m)

(m)

V(d) := {(f, ζ) ∈ P(H(d) ) × Pn (C) : fi (ζ) = 0, 1 ≤ i ≤ m}. (m)

(m)

This subset V(d) ⊆ P(H(d) )×Pn (C) is a complex smooth multihomogeneous algebraic variety (m)

of co-dimension m. Thus, its complex dimension is N(d) +n−m. At every point (f, ζ) ∈ V(d) (m)

the tangent space T(f,ζ) V(d) is given by the following equality: (m) ˙ ∈ Tf P(H(m) ) × Tζ Pn (C) : f˙(ζ) + Tζ f (w) T(f,ζ) V(d) := {(f˙, ζ) ˙ = 0}, (d)

where Tζ f : Tζ Pn (C) −→ Cm is the restriction of the jacobian matrix Df (ζ) to Tζ Pn (C) = ζ ⊥ which is the orthogonal complement of ζ in Cn+1 with respect to the canonical Hermitian product in Cn+1 . Namely, we have: Tζ f := Df (ζ)|ζ ⊥ . We have two canonical projections defined in the solution variety: (m)

(m)

(m)

π1 : V(d) −→ P(H(d) ), π2 : V(d) −→ Pn (C). (f, ζ) 7−→ f (f, ζ) 7−→ ζ The following proposition resumes the main properties of these two canonical projections. Proposition 2.2.1 (cf. [BCSS, 98] and [BP, 08]). With the same notations as above, assume 1 ≤ m ≤ n. Then, the following properties hold: (m)

(1) The mapping π2 : V(d) −→ Pn (C) is an onto mapping, a submersion at every point (m)

(f, ζ) ∈ V(d) and for every ζ ∈ Pn (C) the fiber π2−1 ({ζ}) can be identified with a (m)

complex projective linear submanifold of P(H(d) ) of co-dimension m given by the following equality: (m)

Vζ := π2−1 ({ζ}) = {f ∈ P(H(d) ) : fi (ζ) = 0, 1 ≤ i ≤ m}. (m)

(m)

(m)

(2) The mapping π1 : V(d) −→ P(H(d) ) is an onto mapping and for every f ∈ P(H(d) ) the following equality holds: VP (f ) := π1−1 ({f }). (m)

(3) The set of critical values Σ(d) ⊆ P(H(d) ) of π2 is a proper non-empty complex (m)

subvariety of P(H(d) ). It is called the discriminant variety. (m)

(4) For every system f ∈ P(H(d) r Σ(d) , outside the discriminant variety, the fiber VP = π1−1 ({f }) is an smooth complete intersection complex projective subvariety of co-dimension m. 2.2.1.2. The over-determined case. Things are somehow different if we consider the overdetermined case (i.e. m ≥ n + 1). Here, we just consider the case m = n + 1 which may be resumed in the following statement. Proposition 2.2.2 (cf. [CD, 05] and its references, for instance). With the same notations as in the previous subsection, assume m = n + 1. Then, the following properties hold: (n+1)

(1) The variety V(d) (n+1) P(H(d) )

is an smooth Riemmanian manifold of co-dimension n + 1 in

× Pn (C). (n+1)

(n+1)

(2) The image R(d) := π1 (V(d) ) is a projective variety of co-dimension one (and, hence, a complex hyper-surface).

22

2. BASIC NOTIONS (n+1)

(3) The variety R(d) is defined as the set of common zeros of an irreducible (and hence primitive) polynomial (n+1)

Res(d)

∈ Z[

n [

n+1 {A(i) , |µ| = di }], µ : µ∈N

i=0 (i)

which is multi–homogeneous on each group of variables {Aµ (n+1) ing the coefficients of the polynomials in Hdi .

: |µ| = di } represent(i)

(4) Moreover, the degree on Resn+1 (d) on each group of variables {Aµ Q j6=i dj . P Q (n+1) (5) The total degree of Res(d) is ni=1 j6=i dj . (n+1)

The polynomial Res(d) (n+1)

variety R(d)

(n+1)

⊆ P(H(d)

: |µ| = di } equals

) is usually called the multi-variate resultant and the

is usually called the (multi-variate) resultant variety. (m)

2.2.2. A few words about Probability Distributions on H(d) . The complex projective space Pn (C) is a complex Riemannian manifold with the metric induced by the canonical Hermitian product h·, ·i : Cn+1 × Cn+1 → Cn . This Riemannian structure is associated to a volume form on Pn (C) which will be denoted by dνP . The total volume of Pn (C) with respect to this volume form dνP is finite and it is given by the following equality: πn πn νP [Pn (C)] = = , Γ(n + 1) n! where Γ is Riemannian’s gamma function. Thus, we may define a probability distribution on Pn (C) in the following terms: For every integrable function f : Pn (C) → R, we define its expectation by the following equality: Z 1 f (z)dνP (z). Ez∈Pn (C) [f ] := νP [Pn (C)] Pn (C) Similarly, the complex sphere S 2n+1 := {z ∈ Cn+1 : kzk2 = 1} has a natural Riemannian structure induced by the Hermitian product on Cn+1 and a volume form that we denote by dνS . Again, the volume of S 2n+1 is finite and it is given by the following equality: νS [S 2n+1 ] :=

2π n+1 . Γ(n + 1)

This also allows us to define a probability distribution on S 2n+1 (usually called the “uniform” distribution on S 2n+1 ) such that for every integrable function f : S 2n+1 → R its expectation is given by the following equality: Z 1 Ez∈S 2n+1 [f ] := f (z)dνS (z). νS [S 2n+1 ] S 2n+1 Finally, identifying Cn+1 ∼ = R2(n+1) we may also consider de Gaussian probability distribution which is, as usual, defined by the following terms: For every integrable function f : Cn+1 → R its expectation with respect to the Gaussian distribution is given by the following equality: Z kzk2 1 − 2 Ez∈γ 2n+2 [f ] := f (z)e dz. (2π)n+1 Cn The following statement relates these three probability distributions. Proposition 2.2.3. Let f : Cn+1 → R be homogeneous of degree 0 (i.e. for all λ ∈ C r {0}, f (λz) = f (z)). Let us define the following functions: fS : S 2n+1 −→ R, fP : Pn (C) −→ R. z 7−→ f (z) z 7−→ f (z)


23

Then, f is integrable with respect to the Gaussian distribution if and only if any of the functions fS or fP are integrable with respect to dνS and dνP respectively. Moreover, the following equalities hold among the respective expectations: Ez∈γ 2n+2 [f ] = Ez∈S [fS ] = Ez∈Pn (C) [fP ]. Proof.– For a proof of the first equality see [BCSS, 98], Prop. 1, p. 208. It simply follow by integrating in spherical coordinates. As for the second property simply observe that the following equality holds: Z Z 1 fP (z)dνP (z). fS (z)dνS (z) = 2π S 2n+1 Pn (C) (m)

A similar discussion can be done with the space of polynomial equations H(d) , although we replace the canonical Hermitian product by Bombieri-Weyl’s one. Thus, we also denote by (m) (m) dνP the volume form on P(H(d) ) induced by the complex Riemannian structure on P(H(d) ) (m)

induced by Bombieri-Weyl’s Hermitian product h·, ·i∆ on H(d) . We have (m)

νP [P(H(d) )] =

πN , Γ(N + 1) (m)

where N = N(d) is the complex dimension of P(H(d) ). We also have the expectation of every (m)

integrable function ϕ : P(H(d) → R given by the following equality: Z 1 ϕ(f )dνP (f ). Ef ∈P(H(m) ) [ϕ] = (m) (m) (d) ) νP [P(H(d) )] P(H(d) (m)

In what follows we may either consider the complex sphere S2N +1 = S(H(d) ) with respect to Bombieri-Weyl’s norm or the product of spheres m Y (m) (n+1) S(d) := S(Hdi ). i=1

We respectively have the volume forms dνS and dνS associated to these Riemannian structures. The total volumes are also finite and satisfy: (m)

νS [S(H(d) ] = and (m)

νS [S(d) ] =

2π N +1 , Γ(N + 1)

m Y 2π Ni +1 , Γ(Ni + 1) i=1

m X di + n (n+1) where Ni + 1 = is the complex dimension of Hdi and N + 1 = Ni + 1 is the n i=1

complex dimension of ϕ:

S2N +1

→ R and ψ

(m) H(d) . (m) : S(d)

We may similarly define the expectation of integrable functions

→ R by the following identities: Z 1 ϕ(f )dνS (f ), Ef ∈S(H(m) ) [ϕ] := νS [S2N +1 ] S2N +1 (d) Z 1 Ef ∈S(m) [ψ] := ψ(f )dνS (f ). (m) (d) νS [S(d) ] S(m) (d) (m)

2N +2 We may finally consider the Gaussian distribution γ∆ on C2N +2 = H(d) , defined by (m)

Bombieri-Weyl’s norm. Namely, for every measurable function ϕ : H(d) → R we define its Gaussian equation Z kf k2 1 − 2∆ Ef ∈γ 2N +2 [ϕ] := ϕ(f )e df, ∆ (2π)N +1 C2N +2

24

2. BASIC NOTIONS

where kf k2∆ is the square of the Bombieri-Weyl’s norm of f . The following Proposition mimics the previous one involving distributions induced by the canonical Hermitian product in Cn+1 . (m) A function ϕ : H(d) → R is called m-multi-homogeneous of degree zero if for all λ = (m)

(λ1 , . . . , λm ) ∈ Cm , λi 6= 0, 1 ≤ i ≤ m, and for all f = (f1 , . . . , fm ) ∈ H(d) , fi 6= 0, 1 ≤ i ≤ m, the following equality holds: ϕ(λ · f ) := ϕ(λ1 f1 , . . . , λm fm ) = ϕ(f1 , . . . , fm ). Note that if ϕ is m-multi-homogeneous of degree zero it is also homogeneous of degree zero outside the following set: ! ! m i−1 m [ Y Y (n+1) (n+1) Hdk × {0} × Hdk . i=1

k=1

k=i+1

(m)

Proposition 2.2.4. Let ϕ : H(d) → R be an m-multi-homogeneous function of degree zero. Let us define the functions ϕS := ϕ|S(H(m) ) , ϕS := ϕ|S(m) , (d)

and

(d)

(m)

ϕP : P(H(d) ) −→ R, z 7 → ϕ(π −1 (z)) − (m)

(m)

where π : H(d) r {0} → P(H(d) ) is the canonical projection onto this complex projective space. 2N +2 if and only if any of the Then, ϕ is integrable with respect to the Gaussian distribution γ∆ functions ϕS , ϕS or ϕP is integrable respectively with respect to dνS , dνS or dνP . Moreover, the following equalities hold: Eγ 2N +2 [ϕ] = ES2N +1 [ϕS ] = ES(m) [ϕ] = EP(H(m) ) [ϕ]. ∆

(d)

(d)

2N +2 , and our From now on we will make no reference to Gaussian distribution γ 2N +2 or γ∆ study will just consider either of the expectations ES , ES or EP .

2.2.3. Normal Jacobians and the Co-area Formula. The Co-area Formula is a classic integral formula which generalizes Fubini’s Theorem. The most general version we know is Federer’s Co-area Formula (cf. [Fe, 69]), but for our purposes a smooth version as used in [BCSS, 98] and references therein, or [Ho, 94] suffices. Definition 2.2.1. Let X and Y be Riemannian manifolds, and let F : X −→ Y be a C 1 surjective map. Let k = dim(Y ) be the real dimension of Y . For every point x ∈ X such that the differential DF (x) is surjective, let v1x , . . . , vkx be an orthonormal basis of Ker(DF (x))⊥ . Then, we define the Normal Jacobian of F at x, N Jx F , as the volume in the tangent space TF (x) Y of the parallelepiped spanned by DF (x)(v1x ), . . . , DF (x)(vkx ). In the case that DF (x) is not surjective, we define N Jx F = 0. The following Proposition is easy to prove from this Definition. Proposition 2.2.5. Let X, Y be two Riemannian manifolds, and let F : X −→ Y be a C 1 map. Let x1 , x2 ∈ X be two points. Assume that there exist isometries ϕX : X −→ X and ϕY : Y −→ Y such that ϕX (x1 ) = x2 , and F ◦ ϕX = ϕY ◦ F. Then, the following equality holds: N Jx1 F = N Jx2 F. Moreover, if there exists an inverse G : Y −→ X, then 1 N Jx F = . N JF (x) G


25

A relevant tool to be used in forthcoming pages in the following classical statement of Integral Geometry: Theorem 2.2.6 (Co–area Formula). Consider a differentiable map F : X −→ Y, where X, Y are Riemannian manifolds of real dimensions n1 ≥ n2 . Consider a measurable function f : X −→ R, such that f is integrable. Then, for every y ∈ Y except a zero–measure set, F −1 (y) is empty or a real submanifold of X of real dimension n1 −n2 . Moreover, the following equality holds (and the integrals appearing on it are well defined): Z Z Z f (x) dF −1 (y)dY. f N Jx F dX = y∈Y

X

x∈F −1 (y)

The following statement is Lemma 21 of [BP, 07]. Lemma 2.2.7. Let ϕ0 : Cn −→ Pn (C) be the canonical embedding introduced in Equation (1.2.1) in Chapter 1 above. Then, the following equality holds for every z ∈ Cn : N Jz ϕ0 :=

1 (1 + ||z||2 )n+1

.

In particular, for every f ∈ C[X1 , . . . , Xn ], the following inequality holds: Z Z log |f (z1 , . . . , zn )| log |f (ϕ−1 0 (x))|dνP (x), n+1 dz = 2 n ) (1 + ||z|| x∈Pn (C) z∈C where dνP is the canonical form associated with the Fubini–Study metric in Cn+1 . 2.2.4. Double fibration and some integral formulae. As in [BP, 08], the following statement will be extensively used in forthcoming pages. Let (d) := (d1 , . . . , dn ) be a degree (n) list. We may introduce a variation of the solution variety V(d) introduced in previous pages. (n)

(n)

We define the incidence variety V(d) := V(d) ⊆ S(d) × Pn (C) in the following terms: (n)

(n)

V(d) = V(d) := {(f, ζ) ∈ S(d) × Pn (C) : fi (ζ) = 0, 0 ≤ i ≤ n, f = (f1 , . . . , fn )}. As in previous pages we may also consider two canonical projections: (n)

• π1 : V(d) −→ S(d) , π1 (f, ζ) := f, ∀(f, ζ) ∈ V(d) . • π2 : V(d) −→ Pn (C), π2 (f, ζ) := ζ, ∀(f, ζ) ∈ V(d) . From Proposition 2.1.2 there is an isometric action of the unitary group U(n + 1) on V(d) given in the following terms: U(n + 1) × V(d) −→ V(d) ∗ (U, (f, ζ)) 7−→ (f ◦ U , U ζ).

(2.2.1)

The following double fibration argument is a well–known statement and technique, used in different forms in former manuscripts (cf. [BP, 09a], [BoP, 08] and their references, for instance). We just include it for shake of completeness and the calculation of the involved constants. Proposition 2.2.8. Let g : Pn (C) −→ R+ be an integrable function. Then, the following equality holds:   (n) Z X D(d) νS [S(d) ] Z   g(ζ) dνS (f ) = g(z)dνP (z), (n) νP [Pn (C)] Pn (C) f ∈S ζ∈VP (f )

(d)

where dνP is the differential form associated to the canonical Fubini-Study metric in Pn (C). Proof.– Note that applying the Co-area Formula with respect to π1 we have:   Z Z X   I(g) := g(ζ) dνS (f ) = g(ζ)N J(f,ζ) π1 dV(d) (f, ζ). (n)

f ∈S(d)

ζ∈VP (f )

(f,ζ)∈V(d)

26

2. BASIC NOTIONS

Now, we apply the Co–area Formula with respect to the projection π2 to conclude: ! Z Z N J(f,z) π1 −1 g(z) I(g) := dπ (z)(f, z) dνP (z). N J(f,z) π1 2 z∈Pn (C) π2−1 (z) Now, due to the isometric action of U(n + 1) on V(d) , we conclude that the following quantity: ! Z N J(f,z) π1 −1 T := Tz := dπ2 (z)(f, z) , π2−1 (z) N J(f,z) π1 is constant and independent of z. Moreover, we have ! Z I(g) := T g(z)dνP (z) . z∈Pn (C)

Q Taking g ≡ 1 in this identity and noting that ] π1−1 (f ) = D(d) = ni=1 di holds almost every (n)

(n)

where in P(H(d) ), we conclude that I(1) = D(d) νS [S(d) ] and, hence, (n)

D(d) νS [S(d) ] = νP [Pn (C)]T. We thus conclude:

(n)

T := and hence

D(d) νS [S(d) ] νP [Pn (C)]

,

(n)

I(g) =

D(d) νS [S(d) ] Z νP [Pn (C)]

Pn (C)

g(z)dνP (z),

and the claim follows.

Corollary 2.2.9. With the previous notations and assumptions, this Proposition may be rephrased:   X ES(n)  g(ζ) = D(d) EPn (C) [g] . (d)

ζ∈VP (f )

Analogously to the previous Proposition, using unitary invariance, the following statements hold for the diverse distributions we introduced in previous subsections. Corollary 2.2.10. Let g : Pn (C) → R be an integrable function, 1 ≤ m ≤ a positive integer and (d) = (d1 , . . . , dm ) a list of degrees. Then, the following equalities hold: "Z # Ef ∈S(H(m) ) (d)

VP (f )

"Z Ef ∈P(H(m) ) (d)

where D(d) =

Qm

i=1 di

VP (f )

g(ζ)dVP (f ) = D(d) Ez∈Pn (C) [g(z)], # g(ζ)dVP (f ) = D(d) Ez∈Pn (C) [g(z)],

is the Bézout number associated to the degree list (d) = (d1 , . . . , dm ).

CHAPTER 3

Proof of the Main outcomes about expected distances and separations 3.1. Proofs of the statements about expectations, distances and separation bounds In this section we prove Theorems 1.3.5, 1.3.6, 1.3.8, 1.3.9, as stated in Subsection 1.3.4. 3.1.1. Proof of Theorem 1.3.5. Now we are going to prove Theorem 1.3.5 above. We are going to use a technique similar to the double fibration technique described in Subsection 2.2.4 above. Proof.– So, we are interested to compute the value of the following integral: Z δav (f, g)dνP (f )dνP (g). I := (s)

(m)

P(H(d) )×P(H(d0 ) )

With the same notations of Subsection 2.2.4 above, let us consider the incidence variety (m) (s) V(d)⊕(d0 ) := V(d) × V(d0 ) . As in the case of the solution variety, we have two canonical projections: (m)

(s)

π1 : V(d)⊕(d0 ) −→ P(H(d) ) × P(H(d) ), π2 : V(d)⊕(d0 ) −→ Pn (C) × Pn (C). m ) × P(Hs ) we have Note that for (f, g) ∈ P(H(d) (d0 )

π1−1 (f, g) = VP (f ) × VP (g), and for (x, y) ∈ Pn (C) × Pn (C) we have π2 −1({(x, y)}) = Vx(m) × Vy(s) , where (m)

:= {f ∈ P(H(d) ) : fi (x) = 0, 1 ≤ i ≤ m, f = (f1 , . . . , fm )},

(s)

:=

Vx Vy

(m)

(s)

{g ∈ P(H(d) ) : gj (y) = 0, 1 ≤ j ≤ s, g = (g1 , . . . , gs )}.

Note that generically, the following equalities hold: vol[VP (f )] = D(d) νn−m , vol[VP (g)] = D(d0 ) νn−s , where D(d) and D(d0 ) are the respective Bézout numbers associated to the degree lists (d) and (d0 ), and νk is the volume of the projective space Pk (C). Thus, we have Z 1 dP (x, y)N J((f,x),(g,y)) π1 dV(d)⊕(d0 ) . I := D(d) νn−m D(d0 ) νn−s V(d)⊕(d0 ) Additionally I :=

1 D(d) νn−m D(d0 ) νn−s

Z Pn (C)2

Z (m)

Vx

(s)

×Vy

dP (x, y)

N J((f,x),(g,y)) π1 (m) dV (f )dVy(s) (g)dνP (x)dνP (y). N J((f,x),(g,y)) π2 x

Now, let us consider the following isometric action of U(n + 1)2 on V(d)⊕(d0 ) : U(n + 1)2 × V(d)⊕(d0 ) −→ V(d)⊕(d0 ) . ((U, V ), ((f, ζ), (g, ζ 0 ))) 7−→ ((f ◦ U ∗ , U ζ), (g ◦ V ∗ , V ζ 0 )) 27

28

3. PROOF OF THE MAIN OUTCOMES ABOUT EXPECTED DISTANCES AND SEPARATIONS

This isometric action preserves normal jacobians and integrals. Then, as in the proof of the double fibration technique (cf. Subsection 2.2.4), the following quantity is constant and independent of (x, y) ∈ Pn (C)2 : Z N J((f,x),(g,y)) π1 (m) 1 Jx,y := dV (f )dVy(s) (g). D(d) νn−m D(d0 ) νn−s Vx(m) ×Vy(s) N J((f,x),(g,y)) π2 x Moreover, as in the proof of the double fibration technique again, observe that: Z Z (m) (s) (m) (s) 1dP(H(d) )dP(H(d0 ) ) = νP [P(H(d) )]νP [P(H(d0 ) )]. Jx,y dxdy = Pn (C)2

(m)

(s)

P(H(d) )×P(H(d0 ) )

Hence, we conclude that for all (x, y) ∈ Pn (C)2 we have: (m)

Jx,y = and, hence Z I := (x,y)∈Pn (C)2

(s)

νP [P(H(d) )]νP [P(H(d) )] νP [Pn (C)2 ]

,

(m)

(s)

dP (x, y)Jx,y dνP (x)dνP (y) = νP [P(H(d) )]νP [P(H(d) )]EPn (C)2 [dP (x, y)].

Namely, we have proved that EP(H(m) )×P(H(s) ) [δav (f, g)] = EPn (C)2 [dP (x, y)], (d0 )

(d)

and the statement of this Theorem 1.3.5 is an immediate consequence of the following Lemma. Lemma 3.1.1. With the same notations as above, EPn (C)2 [dP (x, y)] = 1 −

1 . 2n + 1

Proof.– According to Fubini and Study, the following equality holds (cf [Ch, 99], for instance): νP [BP (x, ε)] = νP [Pn (C)]ε2n , where x ∈ Pn (C), ε ∈ R, ε > 0 is a positive real number, BP (x, ε) is the closed ball of radius ε centered at x with respect to the projective distance dP . Now, observe that for every fixed y ∈ Pn (C), the following quantity is constant and independent of y: Ey := Ex∈Pn (C) [dP (·, y)]. This is constant since this quantity is invariant under the action of the unitary group U(n+1) (as it is dP (·, y)) and, hence, we have: EPn (C)2 [dP (x, y)] = Ey∈Pn (C) Ex∈Pn (C) [dP (x, y)] = Ey∈Pn (C) [Ey ] = Ey , for any y ∈ Pn (C). In order to conclude the proof of the Lemma, simply observe that Z ∞ Ey := ProbPn (C) [x : dP (x, y) ≥ t]dt, 0

where νP [x : dP (x, y) ≥ t] . νP [Pn (C)] As for t > 1 the set of points x ∈ Pn (C) such that dP (x, y) ≥ t is the empty set, we conclude: Z 1 Ey := ProbPn (C) [x : dP (x, y) ≥ t]dt. ProbPn (C) [x : dP (x, y) ≥ t] :=

0

This quantity, obviously equals: Z

1

1−

Ey := 0

νP [BP (y, t)] νP [Pn (C)]

dt.

3.1. PROOFS OF THE STATEMENTS

Thus, using the previous equality, this yields: Z 1 Ey := 1 − t2n dt = 1 − 0

29

1 , 2n + 1

and the Lemma follows.

3.1.2. Proof of Theorem 1.3.6. First of all, recall the following statement of [BP, 07]: Theorem 3.1.2 ([BP, 07]). Let V ⊆ Pn (C) be an equi–dimensional complex algebraic variety. Let Vε be the tube about V of radius ε with respect to the distance dP . Namely, let Vε ⊆ Pn (C) the subset given by the following identity: Vε := {y ∈ Pn (C) : dP (V, y) ≤ ε}. Then, we have eε 2r νP [Vε ] ≤ 2deg(V ) , νP [Pn (C)] r where deg(V ) is the geometric degree of V ⊆ Pn (C), νP [Vε ] is the volume in Pn (C) of the tube Vε and r is the co-dimension of V in Pn (C). Now we are going to prove Theorem 1.3.6 above. Proof.– According to Macaulay’s Unmixedness Theorem, as VP (g) is given by s equations and its dimension is n − s, then it is an equi-dimensional algebraic variety and eε 2s νn [VP (g)ε ] ≤ 2νn [Pn (C)]D(d0 ) . s (n)

As m = n, s ≥ 1 and m + s > n, VP (g) ∩ VP (f ) = ∅ for almost all f ∈ P(H(d) ). Thus, it has sense to consider the following integral Z Z 1 1 I := dνP (f ) = max : ζ ∈ VP (f ) dνP (f ). (n) (n) dP (ζ, VP (g)) f ∈P(H ) dP (VP (f ), VP (g)) f ∈P(H ) (d)

(d)

(n)

Note that this equality holds because VP (f ) ⊆ Pn (C) is a finite set for almost all f ∈ P(H(d) ). Moreover, we have Z X 1 I ≤ J := df. (n) d (ζ, VP (g)) P f ∈P(H ) (d)

ζ∈VP (f )

Using the double fibration technique as in Subsection 2.2.4 above, we conclude (n) D(d) νP [P(H(d) )] Z 1 (3.1.1) J= dνP (z). νP [Pn (C)] d (z, VP (g)) Pn (C) P Now, observe that the following classical property holds for every positive function on a probability space: Z 1 1 1 EPn (C) = dνP (z) = dP (·, VP (g)) νP [Pn (C)] Pn (C) dP (z, VP (g)) Z ∞ 1 = ProbP ≥ t dt, dP (z, VP (g)) 0 where ProbP [A] is the probability of subset A ⊆ Pn (C) with respect to the chosen probability distribution. Now, observe that 1 νP [VP (g)t−1 ] ≥t = . ProbP dP (z, VP (g)) νP [Pn (C)] Observe that for t < 1, VP (g)t−1 = Pn (C) and, hence, we have Z ∞ 1 1 EPn (C) =1+ νP [VP (g)t−1 ]dt. dP (·, VP (g)) νP [Pn (C)] 1

30


Using the bounds of Theorem 3.1.2 above we conclude: Z ∞ e 2s dt 1 EPn (C) ≤1+ 2 deg(VP (g)) , dP (·, VP (g)) s t2s 1 and, hence, we have EPn (C)

e 2s 1 1 ≤ 1 + 2 deg(VP (g)) . dP (·, VP (g)) s 2s − 1

Replacing this inequality in 3.1.1 above, we conclude e 2s 1 (n) J ≤ D(d) νP [P(H(d) )] 1 + 2 deg(VP (g)) . s 2s − 1 This yields Ef ∈P(H(n) ) (d)

e 2s 1 1 ≤ D(d) 1 + 2 deg(VP (g)) . dP (VP (f ), VP (g)) s 2s − 1

Using Jensen’s Inequality, we conclude Ef ∈P(H(n) ) [dP (VP (f ), VP (g))] ≥ (d)

=

s2s (2s − 1) = D(d) s2n (2s − 1) + 2 deg(VP (g))D(d) e2s 2s − 1

D(d) + 2 deg(VP (g))D(d)

. e 2s s

This proves the inequality in equation 1.3.3 of Theorem 1.3.6. The remaining inequalities s Y (s) follow since deg(VP (g)) = d0i for almost all g ∈ P(H(d) ). i=1

3.1.3. Proof of Theorem 1.3.8. This proof uses Proposition 22 in [BP, 11a]. First, of all, observe that the following equalities hold: X 1 sepav (f ) := dP (ζ, ζ 0 ) = D(d) (D(d) − 1) 0 0 ζ,ζ ∈VP (f ),ζ6=ζ

 =

1 D(d)

X  ζ∈VP (f )

 1 (D(d) − 1)

X

dP (ζ, ζ 0 ) .

ζ 0 ∈VP (f ),ζ 0 6=ζ

Then, we have 1 sepav (f ) ≥ D(d)

1 

ζ∈VP (f )

(D(d) − 1)

√ 3− 7 3− 7 X 1 = , 2γ(f, ζ) D(d) 2γ(f, ζ) √

 X

X ζ 0 ∈VP (f ),ζ 0 6=ζ



ζ∈VP (f )

where γ(f, ζ) is the quantity introduced in [SS, 93], Now, using the “higher derivative estimate” of [BCSS, 98], we conclude: √ 1 3− 7 X sepav (f ) ≥ µnorm (f, ζ)−1 , 3/2 D(d) d ζ∈VP (f )

where µnorm (f, ζ) is the non-linear condition number introduced in [SS, 93]. Then, we have   √ X 3− 7 1 EP(H(n) ) [sepav ] ≥ EP(H(n) )  µnorm (f, ζ)−1  . 3/2 D d (d) (d) (d) ζ∈VP (f )

Now, using Proposition 22 of [BP, 11a], we get √ 3 − 7 Γ(N + 1) Γ(n2 + n + 1/2) EP(H(n) ) [sepav ] ≥ EM ∈S(H(n) ) [||M † ||−1 ], 2 3/2 Γ(N + 3/2) Γ(n + n) d (d) (1)

3.1. PROOFS OF THE STATEMENTS

31

(n)

(n)

where H(1) is the space of matrices M which define linear mappings M : Cn+1 → Cn , S(H(1) ) (n)

is the sphere of radius 1 in H(1) with respect to Frobenius norm, M † is Moore-Penrose pseudoinverse and ||M † ||−1 is the inverse of the norm of M † as linear operator. Using Gautschi and Kershaw Inequalities (cf. [Ga, 59], [Kr, 83]) we get √ p 3 − 7 N + 1/4 Γ(n2 + n + 1/2) EP(H(n) ) [sepav ] ≥ EM ∈S(H(n) ) [||M † ||−1 ]. Γ(n2 + n) d3/2 N + 1/2 (d) (1) And also

√ 1/2 3− 7 Γ(n2 + n + 1/2) 2 p EP(H(n) ) [sepav ] ≥ EM ∈S(H(n) ) [||M † ||−1 ]. 2 3/2 3 Γ(n + n) (d) (1) d N + 1/2

Now we use Jensen’s inequality to conclude √ 1/2 3− 7 Γ(n2 + n + 1/2) 1 2 p EP(H(n) ) [sepav ] ≥ . 2 3 Γ(n + n) EM ∈S(H(n) ) [||M † ||] (d) d3/2 N + 1/2 (1)

Using Theorem 19 of [BP, 11a], we have: √ 1/2 2 Γ(n2 + n + 1/2) 3 3− 7 p EP(H(n) ) [sepav ] ≥ . 3 Γ(n2 + n) 8n3/2 (d) d3/2 N + 1/2 Using Gautschi again, we have: √ 1/2 2 3(n2 + n − 1/4)1/2 3− 7 p EP(H(n) ) [sepav ] ≥ . 3 8n3/2 (d) d3/2 N + 1/2 This finally gives

√ EP(H(n) ) [sepav ] ≥ (d)

6(3 − 8

√ s 7) d3 (N

1 . + 1/2)n

And this proves Theorem 1.3.8.

3.1.4. Proof of Theorem 1.3.9. With the same notations and citations of the previous proof, note that the following equalities and inequalities hold: 1 2γ(f, ζ) 1 0 √ : ζ ∈ VP (f ) . = max : ζ, ζ ∈ VP (f ) ≤ max sepmin (f ) dP (ζ, ζ 0 ) 3− 7 Also as in the previous proof, using the “higher derivative estimate” as above, we have: X d3/2 µnorm (f, ζ) 1 √ ≤ . sepmin (f ) 3− 7 ζ∈V (f ) P

Then, we also have: ES(H(n) ) (d)

1 sepmin (f )

≤

d3/2 3−



 X

√ ES(H(n) )  (d) 7 ζ∈V

µnorm (f, ζ) .

P (f )

According to Proposition 22 and Theorem 19 of [BP, 11a], we have:   X X n + 1 Γ(n − k + 1/2) D(d) d3/2 Γ(N + 1) n−1 d3/2   √ E (n) √ µnorm (f, ζ) = . k nn−k+1/2 Γ(n − k) 3 − 7 S(H(d) ) ζ∈V (f ) 3 − 7 Γ(N + 1/2) k=0 P

Using Gautschi and Kershaw Inequalities (cf. [Ga, 59], [Kr, 83] or [GQ, 08] for precise details), we conclude: D(d) d3/2 1 n + 1/2 1/2 1 n+1 √ (N + 1/2)1/2 ES(H(n) ) ≤ 1+ . sepmin (f ) n n (d) 3− 7

32


Roughly, we have: 4eD(d) d3/2 √ (N + 1/2)1/2 , sepmin (f ) (d) 3− 7 where e is the basis of the natural logarithm. Now, from Holder’s inequality, one easily concludes: 4eD(d) d3/2 1 1 √ (N + 1/2)1/2 , ≤ ≤ ES(H(n) ) ES(H(n) ) [sepmin (f )] sepmin (f ) (d) 3− 7

ES(H(n) )

1

≤

(d)

and hence, we conclude the wanted inequality:

√ 3− 7 ES(H(n) ) [sepmin (f )] ≥ (N + 1/2)−1/2 . 4eD(d) d3/2 (d)

CHAPTER 4

Arithmetic Poisson Formula and Applications 4.1. Mahler’s measure of multivariate polynomials Mahler’s measure has been used as an effective measure of the size of polynomials. It has been considered better suited than the canonical norm k·k2 because of its good behaviour with respect to the product of polynomials (namely M (f · g) = M (f ) · M (g)). Bombieri-Weyl’s norm is the only serious competitor since satisfies Bombieri’s inequality (cf. Proposition 2.1.2 below). Nevertheless, Mahler’s measure remains as the main archimedean measure of polynomials with coefficients in a number field and, hence, Mahler’s measure is the main quantity to study in Arithmetic/Diophantine Geometry. The logarithm of the Mahler’s measure is, in fact,Qan expectation of the logarithm of the polynomial along the product of complex circles ni=1 S 1 ⊆ Cn . This is a feature that Mahler’s measure shares with Bombieri-Weyl’s norm. In the early nineties, P. Philippon (cf. [Ph, 91], [Ph, 95]) introduced some variations of the logarithmic Mahler’s measure, replacing the product of circles by products of spheres and even the sphere S 2n−1 ⊆ Cn . Its main outcome in [Ph, 91] was Theorem 4.1.2 below just comparing the values of the logarithmic Mahler’s measures when we choose either the Q product ni=1 S 1 or the sphere S 2n−1 . In this Section we recall the main notations about the Mahler’s measure according to Philippon. We show some technical results concerning the case of polynomials with less variables (Proposition 4.1.5) and a main outcome that we resume here. (n+1) As in previous sections, let Hd0 be the complex vector space of all homogeneous complex (n+1)

polynomials in C[X0 , . . . , Xn ] of degree d0 . Let S(Hd0

(n+1)

) be the unit sphere in Hd0

with

(n+1) Hd0 .

respect to Bombieri-Weyl’s metric. Let R + 1 be the complex dimension of As in Section 1.2 we also denote here by Hk the k-th harmonic number. Then, we state and prove the following technical version of Main Theorem 1.3.4. Theorem 4.1.1. Let ϕ0 : Cn −→ Pn (C) be the standard embedding of Cn into Pn (C) (see Equation (1.2.1) above). Let us define the expectation: E := Ef ∈S(H (n+1) ) [EPn (C) [log |a f ◦ ϕ−1 0 |]]. d0

We have: (4.1.1)

d0 0≤E= 2

HR 1 Hn − ≤ d0 (log(n) + γ) + O . d0 n

4.1.1. Mahler’s measure. Let us recall the definition of Mahler’s measure and let us prove some basic properties. Let (m) := (m1 , . . . , mr ) be a list of positive integer numbers. Suppose (m1 + 1) + · · · + (mr + 1) = n + 1. Let us denote by S(m) be the product of spheres: S(m) :=

r Y

S 2mi +1 ⊆ Cn+1 ,

i=1

where S 2mi +1 := {z ∈ Cmi +1 : kzk22 = 1} is the sphere of radious one in Cmi +1 centered at the origin, and k·k2 is the canonical Hermitian norm in Cn+1 . Definition 4.1.1 (Logarithmic Mahler’s measure, [Ph, 91]). Let f ∈ C[X0 , X1 , . . . , Xn ] be a complex polynomial (not necessarily homogeneous). We define the logarithmic Mahler’s 33

34

4. ARITHMETIC POISSON FORMULA AND APPLICATIONS

measure of f with respect to S(m) as: 1 2mi +1 ] ν i=1 S [S

Z

mS(m) (f ) := Qr

log |f (z)|dνS (z), S(m)

where dνS is the volume form associated to the standard Fubini-Study metric in S(m) . We define the absolute Mahler measure of f with respect to S(m) as: MS(m) (f ) := emS(m) (f ) . Mahler’s measure of a polynomial is usually defined as m(p) := mS (p), where S := The following inequalities hold:

Qn

i= S

1.

Theorem 4.1.2 ([Ph, 91], [Le, 94]). For every polynomial f ∈ C[X1 , . . . , Xn ], the following inequalities hold: mS 2n−1 (f ) ≤ m(f ) ≤ mS 2n−1 (f ) + 4 deg(f ) log(n). 1 mS 2n−1 (f ) ≤ m(f ) ≤ mS 2n−1 (f ) + deg(f ) Hn−1 . 2 The following properties also hold for Mahler’s measure: Proposition 4.1.3. With these notations we have: (1) Absolute Mahler’s measure is multiplicative: MS (f g) = MS (f )MS (g). (n) (2) For every polynomial f ∈ Pd of degree d, we have !1/2 d X √ i + n − 1 −1 ||gi ||2i . MS 2n−1 (f ) ≤ d + 1 n−1 i=0

where g0 , . . . , gd ∈ C[X1 , . . . , Xn ] are the homogeneous components of f (of respective degrees 0, . . . , d), ||gi ||i is Bombieri-Weyl’s norm of gi . (3) If f is monic with respect to some variable, then 1 ≤ M (f ). In particular, If f is a monic polynomial in C[X1 , . . . , Xn ] and if g, h ∈ C[X1 , . . . , Xn ] are such that h = f g, then, we have M (g) ≤ M (h). (n+1) (4) For every Polynomial f ∈ Hd , we have d+n 1 + mS 2n−1 (a f ) + d (log(n + 1) + Hn ) , log ||f ||d ≤ log 2 n where kf kd is the Bombieri–Weyl’s norm of f and Hn is the n−th harmonic number. (5) (Bombieri’s inequality) For homogeneous polynomials f and g of respective degrees d and t we have s d!t! kf kd kQkt ≤ kf gkd+t ≤ kf kd kgkt . (d + t)! Proof.– Most of these claims are either known or elementary consequences of the definitions. Let us give more precise relations among some of these Mahler’s measures in the multihomogeneous case. Let X := {X0 , . . . , Xn } be a group of variables and assume it decomposes as a disjoint union of groups of variables X := Y (1) ∪ · · · ∪ Y (r) , where (i)

Y (i) := {Y0 , . . . , Ym(i)i }. Let f ∈ C[X0 , . . . , Xn ] be a multi-homogeneous polynomial with respect to the groups of variables Y (1) , . . . , Y (r) . Let di be the degree of f with respect to the variables in the group Y (i) . Assume, additionally, that f does not depend on all variables. Assume that for every i, (i) (i) 1 ≤ i ≤ r, the polynomial f only depends on the variables Y0 , . . . , Yì and it does not

4.1. MAHLER’S MEASURE OF MULTIVARIATE POLYNOMIALS (i)

35

(i)

depend on the variables Yì +1 , . . . , Ym1 . Let us consider the list (`) = (`1 , . . . , `r ) and define (`)

the subset T(m) ⊆ Cn+1 given by the following identity: (`) T(m)

:=

r Y

S 2ì +1 × {0}mi −ì .

i=1

Q (`) Note that T(m) can be identified with S(`) = ri=1 S 2ì +1 . Due to our hypothesis, our polynomial f ∈ C[X0 , . . . , Xn ], may be viewed as a multi-homogeneous polynomial in the partition X = Y (1) ∪ . . . ∪ Y (r) and as a multi-homogeneous polynomial in less variables X 0 = X (1) ∪ . . . ∪ X (r) , where (j) (j) X (j) = {Y0 , . . . , Ylj }. Thus we may need to compare the logarithmic Mahler’s measures (mS(m) (f ) and mS(l) (f )) The following statements shows how they are related. In what follows, we sometimes use the following function. Given m, ` ∈ N two positive integers, m > `, we consider the function Z 1 t2(m−`)−1 (1 − t2 )`+1 log(1 − t2 )dt. K(m, `) := 0

Lemma 4.1.4. With these notations the following equality holds: m−` 1 X m−` 1 (4.1.2) K(m, `) = − (−1)k = ∆(m−`) [x−2 ](m + 3), 2 k (m − k + 3)2 k=0

where ∆(n) [f ](z) is the n-th forward difference of f at z. Moreover, the following upper and lower bounds hold for K(m, `): (4.1.3)

1 1 − β(m − ` + 1, ` + 1) ≤ K(m, `) ≤ − β(m − ` + 1, ` + 2), 2 2

where β(x, y) =

Γ(x)Γ(y) Γ(x+y)

is the Beta function.

Proof.– In order to prove the equality (4.1.2), observe that Z 1 1 K(m, `) = − (1 − z)m−` z `+1 log zdz. 2 0 Now, observe that Z 1 m−` 1 X m−` k z m−k+1 log zdz. K(m, `) = − (−1) 2 k 0 k=0

If we take u = 1/z, we have Z 1 Z m−k+1 z log zdz = 0

∞

log u um−k+3

1

du =

1 . (m − k + 3)2

This yields m−` 1 X m−` 1 K(m, `) = − (−1)k . 2 k (m − k + 3)2 k=0

As for the inequalities in Equation (4.1.3) above, we use Mercator series of the logarithm. This series implies that for all t ∈ [0, 1], we have − Then, we have Z − 0

t2 ≤ log(1 − t2 ) ≤ −t2 . 1 − t2

1

t2(m−`)+1 (1 − t2 )` dt ≤

Z 0

1

t2(m−`)−1 (1 − t2 )`+1 log(1 − t2 )dt ,

36


and 1

Z

2(m−`)−1

t

2 `+1

(1 − t )

Z log(1 − t )dt ≤ −

1

2

t2(m−`)+1 (1 − t2 )`+1 dt.

0

0

In other terms, we have proved 1 1 − β(m − ` + 1, ` + 1) ≤ K(m, `) ≤ − β(m − ` + 1, ` + 2), 2 2 and the Lemma follows.

Lemma 4.1.5 (Multi–homogenous case). With these notations and assumptions the following equality holds: mS(m) (f ) = J(m),(`) (mS(`) (f ) + J2 ) , where J(m),(`)

n r n X Y X dj (mj − `j ) dj (mj − `j ) ì + 1 ≤ 1, − = ≤ J2 ≤ − ≤ 0. mi + 1 4(`j + 1) 4(mj + 2) j=1

i=1

j=1

Proof.– Let us consider the following integral: Z T := log |f (y (1) , . . . , y (r) )|dy (1) · · · dy (1) , S(m) (i)

(i)

(`)

where y (i) = (y0 , . . . , ymi ) ∈ S 2mi +1 . Let B(m) be the product of balls (`)

B(m) :=

r Y

BCmi −ì (0, 1),

i=1

where BCmi −ì (0, 1) ⊆ Cmi −ì is the closed ball of radius one in Cmi −ì centered at the origin. For every i, 1 ≤ i ≤ r, let πi : S 2mi +1 −→ BCmi −ì (0, 1) the projection onto the last mi − ì complex coordinates. Let us consider the product map Π :=

r Y

(`)

πi : S(m) −→ B(m) .

i=1

According to Lemma 2, p. 206, of [BCSS, 98], the normal jacobian of Π at every point x = (y (1) , . . . , y (r) ) ∈ S(m) is given by the following identity: ! r Y 1 N Jx Π := 1/2 . 1 − ||π(y (i) )||2 i=1 According to Theorem 2.2.6, we conclude: Z Z T := (`)

z∈B(m)

y∈Π−1 (z)

! log |f (y)| −1 (`) dΠ (z) dB(m) . N Jy Π

(`)

(`)

Now, let us denote z := (z (1) , . . . , z (r) ) ∈ B(m) , y := (y (1) , . . . , y (r) ) ∈ Π−1 (z) ⊆ S(m) and, finally, for every i, 1 ≤ i ≤ r, let us denote y (i) := (x(i) , z (i) ) and by x := (x(1) , . . . , x(r) ) ∈ Qr ì +1 . Using these notations, we observe the following immediate properties: i=1 C • log |f (y)| = log |f (x)| = log |f (x(1) , . . . , x(r) )|, !−1 r r 1/2 Y Y 1 • (N Jx Π)−1 = = 1 − ||z (i) ||2 , 1/2 1 − ||π(y (i) )||2 i=1 i=1 r Y 2ì +1 −1 • Π (z) := S√ , where St2k+1 is the sphere of radius t in Ck+1 . (i) 2 i=1

1−||z

||

4.1. MAHLER’S MEASURE OF MULTIVARIATE POLYNOMIALS

37

Thus, we have: r Y

Z (4.1.4)

T :=

(`) z∈B(m)

 1 − ||z (i) ||2

1/2

 Z

 

i=1

Qr

i=1

 (`) log |f (x)|dΠ−1 (z) dB(m) .

2ì +1

S√

1−||z (i) ||2

Now, for every i, 1 ≤ i ≤ r, we consider the mapping 2ì +1 ϕi : S √

S 2ì +1 p 7 → ( 1 − ||z (i) ||2 )−1 x(i) , −

−→

1−||z (i) ||2

x(i) and the product mapping Φ :=

r Y i=1

ϕi :

r Y

2ì +1 S√

−1

1−||z (i) ||2

i=1

=Π

(z) −→

r Y

S 2ì +1 = S(`) .

i=1

(x(1) , . . . , x(r) )

Obviously, Φ defines a bijection and for every x = Jacobian of Φ is given by the following identity: r q Y N Jx Φ := ( 1 − ||z (i) ||2 )−2ì −1 .

∈ Π−1 (z), the Normal

i=1

Moreover, as f is multi–homogeneous in each group of variables, we also have: Q log |f (x(1) , . . . , x(r) )| = log ri=1 (1 − ||z (i) ||2 )di /2 f (ϕ1 (x (1) ), . . . , ϕr (x(r) )) = Pr di /2 log(1 − ||z (i) ||2 ) + log f (ϕ1 (x(1) ), . . . , ϕr (x(r) )) = = i=1 Pr (i) 2 = i=1 di /2 log(1 − ||z || ) + log |f (Φ(x))| . (`)

We thus consider for every z ∈ B(m) the integral Z log |f (x)|dΠ−1 (z), J(z) := Q 2ì +1 r i=1 S√ 1−||z (i) ||2

and, applying the co-area formula (cf. Theorem 2.2.6), as Φ−1 (s) is always a single point, we have: ! Z Z −1 −1 J(z) = log |f (x)| (N Jx Φ) dΦ (s) dS(`) (`) −1 s∈S x∈Φ (s)    Z r r Y X  J(z) = (1 − ||z (i) ||2 )ì +1/2 dj /2 log(1 − ||z (j) ||2 ) + log |f (s)| dS(`) . s∈S(`)

i=1

j=1

In other words, we have proven:    r r X Y dj /2 log(1 − ||z (j) ||2 ) + mS(`) (f ) . J(z) = (1 − ||z (i) ||2 )ì +1/2 νS [S(`) ]  j=1

i=1

We now replace J(z) in Equation (4.1.4) above, to conclude:    Z r r ì +1 Y X (`) T := 1 − ||z (i) ||2 νS [S(`) ]  dj /2 log(1 − ||z (j) ||2 ) + mS(`) (f ) dB(m) . (`)

z∈B(m) i=1

j=1

Namely, we have: (4.1.5)

T := νS [S(`) ] mS(`) (f )J(m),(`) + J2 ,

where Z J(m),(`) :=

(`) z∈B(m)

r Y i=1

1 − ||z (i) ||2

ì +1

(`)

dB(m) ,

38


and J2 :=

n X

r Y

Z dj /2

(`)

1 − ||z (i) ||2

ì +1

z∈B(m) i=1

j=1

(`) log 1 − ||z (j) ||2 ) dB(m) ≤ 0.

We now integrate in polar coordinates to obtain r Z 1 Z Y 2(mi −ì )−1 2 ì +1 mi −ì −1 dt . J(m),(`) := t (1 − t ) dS S 2(mi −ì )−1

0

i=1

Then, we have J(m),(`) :=

r Y

νS [S

2(mi −ì )−1

]

Z

1

2(mi −ì )−1

t

2 ì +1

(1 − t )

dt .

0

i=1

And, then, J(m),(`)

r 1 Y := r νS [S 2(mi −ì )−1 ]β(mi − ì , ì + 2) . 2 i=1

On the other hand, integrating in polar coordinates, we have: J2 :=

n X dj j=1

2

Kj ,

where Kj :=

r Y

 ! Y νS [S 2(mi −ì )−1 ]  β(mi − ì , ì + 2) K(mj , `j ),

i=1

i6=j

where K(·, ·) is the function defined in Lemma 4.1.4 above. Using the upper and lower bounds of Equation (4.1.3), of Lemma 4.1.4 above, we may introduce the quantities Lj and Uj , given by the following identities:  ! r Y Y 1 Lj := r νS [S 2(mi −ì )−1 ]  β(mi − ì , ì + 2) β(mj − `j + 1, `j + 1), 2 i=1

Uj :=

1 2r

i6=j

 ! r Y Y νS [S 2(mi −ì )−1 ]  β(mi − ì , ì + 2) β(mj − `j + 1, `j + 2). i=1

i6=j

And we have: (4.1.6)

−

n X dj j=1

4

Lj ≤ J 2 ≤ −

n X dj j=1

4

Uj .

As

1 T, νS [S(m) ] we perform the obvious calculations with Gamma and Beta functions to conclude: r Y νS [S(`) ] ì + 1 ≤ 1, J(m),(`) = J(m),(`) = mi + 1 νS [S(m) ] mS(m) (f ) =

i=1

and νS [S(`) ] Lj = νS [S(m) ]

νS [S(`) ] Uj = νS [S(m) ]

mj − `j mj + 1

"Y r

mj − `j mj + 2

"Y r

i=1

ì + 1 mi + 1

#

ì + 1 mi + 1

#

,

whereas i=1

.

Replacing these quantities in Equations (4.1.5) and (4.1.6), we finish the proof.

4.1. MAHLER’S MEASURE OF MULTIVARIATE POLYNOMIALS

39

4.1.2. Proof of Theorem 4.1.1. Proof.– As in Lemma 4.6 of [BP, 09a], we apply Lemma 21 of [BP, 07] to conclude: Z Z 1 log |f (1, z)| E= dz dνS (f ). 2 n+1 (n+1) ) Cn (1 + kzk ) νP [Pn (C)]νS [S(Hd0 )] f ∈S(Hd(n+1) 0 For every z ∈ Cn , let us define E(z) the following quantity: ! Z 1 E(z) := log |f (1, z)|dνS (f ) . (n+1) (n+1) f ∈S(Hd ) νS [S(Hd0 )] 0 Observe that the following holds 1 E= νP [Pn (C)]

Z

1 E(z)dz. (1 + kzk2 )n+1

Cn

Cn ,

Now, for fixed z ∈ there is an unitary matrix U ∈ U(n + 1) such that U (1, z) = ((1 + 2 1/2 n+1 kzk ) , 0, . . . , 0) ∈ C . Moreover, due to the unitary invariance of the Bombieri-Weyl’s norm, the following is an isometry: (n+1)

U ∗ : S(Hd0 f

(n+1)

) −→ S(Hd0 ), 7−→ f ◦ U ∗ .

As f is a homogeneous polynomial of degree d0 , we also have: |f (1, ζ)| = |(f ◦ U ∗ )((1 + kzk2 )1/2 , 0, . . . , 0))| = (1 + kzk2 )d0 /2 |(f ◦ U ∗ )(1, 0, . . . , 0)|. Using this equality plus the isometry defined by U ∗ , we have the following equality for every z ∈ Cn : ! Z 1 d0 log(1 + kzk2 ) + log |f ◦ U ∗ (1, 0, . . . , 0) |dνS (f ) . E(z) = (n+1) (n+1) 2 ) f ∈S(Hd νS [S(Hd0 )] 0 Or, equivalently, !

Z

d0 1 E(z) = log(1 + kzk2 ) + (n+1) 2 νS [S(Hd0 )]

(n+1) ) 0

f ∈S(Hd

log |f (1, 0, . . . , 0) |dνS (f ) .

Thus, we conclude: E := J1 + J2 , where d0 J1 := 2νP [Pn (C)] and

J4 :=

Z

1 J3 := νP [Pn (C)] Z 1 (n+1)

νS [S(Hd0

)]

Cn

J2 := J3 J4 , log(1 + kzk2 ) dz ≥ 0, (1 + kzk2 )n+1

Z Cn

1 dz, (1 + kzk2 )n+1 !

(n+1) ) 0

f ∈S(Hd

log |f (1, 0, . . . , 0) |dνS (f ) .

Integrating in polar coordinates we have: Z ∞Z d0 r2n−1 log(1 + r2 ) J1 := dνS dr, 2νP [Pn (C)] 0 (1 + r2 )n+1 S 2n−1 Z ∞Z r2n−1 1 J3 := dνS dr, 2 n+1 νP [Pn (C)] 0 S 2n−1 (1 + r ) νS [S 2n−1 ] As = 2n, we conclude: νP [Pn (C)] J3 =

νS [S 2n−1 ] 1 νS [S 2n−1 ] 1 β(n, 1) = = 1, νP [Pn (C)] 2 νP [Pn (C)] 2n

40


and d0 νS [S 2n−1 ] H = d0 nH, 2νP [Pn (C)]

J1 = where

∞

Z H := 0

r2n−1 log(1 + r2 ) dr . (1 + r2 )n+1

Now, observe that: 1 2

H=

Z

∞

0

r2(n−1) log(1 + r2 ) 1 2rdr = (1 + r2 )n+1 2 =

Z

1 2

∞

∞ (n−1) t log(1

Z

+ t) dt = (1 + t)n+1

0

(x − 1)n−1 log(x) dx. xn+1

1

Hence, n−1 X

1 H := 2

(−1)k

k=0

1 = 2

! Z ∞ n−1−k n−1 x log(x) dx = xn+1 k 1

n−1 X k=0

! Z ∞ n − 1 log(x) (−1)k dx , k xk+2 1

and 1 H := 2

n−1 X k=0

! 1 n−1 1 (−1) = k (k + 1)2 2n k

n−1 X

k

(−1)

k=0

! 1 n Hn = . k + 1 (k + 1) 2n

We thus conclude Hn Hn = d0 . 2n 2 As for studying J4 , let us introduce for α ∈ {0, 1} the following quantity: J1 := d0 n

J4 (α) :=

!

Z

1 (n+1) νS S[S(Hd0 )]

(n+1) ) 0

f ∈S(Hd

logα |f (1, 0, . . . , 0) |dνS (f ) .

Note that f (1, 0, . . . , 0) = a(d0 ,0,...,0) is the coefficient of the monomial X0d0 in the monomial expansion of f . Note that J4 (0) = 1. Moreover, we may consider the projection π(0) : (n+1)

H(d0 ) −→ C given as the projection of f onto the coefficient of the monomial X0d0 . This yields a mapping, which we represent by the same symbol: (n+1)

π(0) : S(Hd0 f

) −→ BC (0, 1), 7−→ f (1, 0, . . . , 0)

where BC (0, 1) is the closed ball in C with center 0 and radius 1. As observed in [BCSS, 98], Lemma 2, page 206, the normal Jacobian of π(0) is known and it is given by N Jf π(0) = (1 − |π(0) (f )|2 )1/2 . Thus, using the co-area formula, we may rewrite J4 (α) in the following terms: ! Z Z 1 J4 (α) := logα |z|(1 − |z|2 )−1/2 df dz. (n+1) −1 f ∈π(0) (z) νS [S(Hd0 )] z∈BC (0,1) Namely, J4 (α) :=

1 (n+1) νS [S(Hd0 )]

Z z∈BC (0,1)

−1 (1 − |z|2 )−1/2 logα |z|vol[π(0) (z)]dz.

4.2. AN ARITHMETIC POISSON FORMULA

41

−1 As π(0) is an onto projection, the inverse image π(0) (z) can be viewed as the sphere of radius (n+1)

(1 − |z|1 )1/2 in Ve0 := V(1,0,...,0) := {f ∈ Hd0 R=

: f (1, 0, . . . , 0) = 0}. Now, let us write

(n+1) dimC (Hd0 )

− 1 to denote the complex dimension of Ve0 and we have: Z νS [S(Ve0 )] J4 (α) = (1 − |z|2 )R−1 logα |z|dz, (n+1) νS [S(Hd0 )] z∈BC (0,1)

where S(Ve0 ) is the sphere of radius one centered at the original in Ve0 . Now, we integrate in polar coordinates in BC (0, 1) to conclude: Z 1Z νS [S(Ve0 )] J4 (α) = (1 − |rθ|2 )R−1 |rθ| logα |rθ|dθdr. (n+1) 1 νS [S(Hd0 )] 0 S Namely, J4 (α) =

νS [S(Ve0 )]νS [S 1 ] (n+1) νS [S(Hd0 )]

Z

1

(1 − r2 )R−1 r logα (r)dr.

0

Taking α = 0 we have: 1 = J4 (0) =

νS [S(Ve0 )]νS [S 1 ] (n+1)

νS [S(Hd0

)]

Z

1

(1 − r2 )R−1 rdr =

0

νS [S(Ve0 )]νS [S 1 ] β(R, 1) , (n+1) 2 νS [S(H )] d0

and, hence, νS [S(Ve0 )]νS [S 1 ] (n+1) νS [S(Hd0 )]

=

2 = 2R. β(R, 1)

Thus, we conclude: 1

Z

(1 − r2 )R−1 r log(r)dr =

J4 = J4 (1) = 2R 0

1 2

Z

1

R(1 − t)R−1 log(t)dt = −

0

HR . 2

4.2. An Arithmetic Poisson Formula Two are the main invariants that measure the size of a polynomial with integer coefficients: its degree and the logarithmic height of its coefficients. When we consider polynomials with coefficients in a number field the two quantities are also considered. The degree keeps the same formulation, whereas the height combines both the canonical archimedean absolute value and also all non-archimedean metrics. In the case of algebraic varieties both measures are generalized using similar terms (as “degree of a variety” or “height of a variety”), although with more sophisticated definitions, which we now resume. The degree of an irreducible variety V of dimension r is defined as the maximum number of intersection points ](V ∩ L) with a generically chosen linear variety L of co-dimension r. The degree of an algebraic variety deg(V ) is defined as the sum of the degrees of its irreducible components. A paramount result was simultaneous and independently stated in the early eighties by J. Heintz (cf. [He, 83]), W. Vogel (cf. [Vo, 84]) and W. Fulton (cf. [Fu, 84]): the Bézout Inequality, which is given by the following inequality: deg(V ∩ W ) ≤ deg(V ) · deg(W ), for any two algebraic varieties. Mimicking this notion from Intersection Theory, several authors developed a notion of height of a diophantine algebraic variety aiming to get, at the end of their studies, an Arithmetic Bézout Inequality. Among the most relevant contributions in this framework we may cite [Ph, 86] [Ph, 91], [Ph, 95],[BGS, 94] in the early and middle nineties. In more recent times, we may also cite [MK, 01], [Re, 01a], [Re, 01b] or [DAKrSo, 12], and references therein.

42


As an example of an arithmetic Bézout inequality, we may outline the main outcome of [Ph, 95] or [BGS, 94]: h(V ∩ W ) ≤ h(V ) deg(W ) + deg(V )h(W ) + c deg(V ) deg(W ), where h(X) is Philippon’s height in [Ph, 95], deg(X) is the degree and   dim(V ) dim(W ) X X dim(V ) + dim(W ) 1 + n− log 2. c= 2(i + j + 1) 2 i=0

j=0

Using these ideas, in [KPS, 01], the authors proved the following statement: Let V ⊆ An be a variety of dimension r, and let ϕ : An → AN be a linear affine map. Then, h(ϕ(V )) ≤ h(V ) + (r + 1)(h(ϕ) + 8 log(n + N + 1)) deg(V ). In the case of projections π : An × Am → An the authors proved (Lemma 2.6 in [KPS, 01]): h(π(V )) ≤ h(V ) + 3(r + 1) log(n + m + 1) deg(V ). Using the Arithmetic Bézout Theorem of [Ph, 95] and its norm of height of this author, in [KPS, 01], the authors proved that for a projection π of a variety V of dimension r, the following inequality holds h(π(V )) ≤ h(V ) + 3(r + 1) log(N + 1) deg(V ),

(4.2.1)

where N is the dimension of the ambient space, r is the dimension of V and π(V ) is the smallest algebraic variety containing π(V ) (cf. Lemma 2.6 in [KPS, 01]). One could try to use this kind of bounds to know the height of the multi-variate resultant (n+1) variety R(d) introduced in Section 2.2 above. Using, for instance, the inequality of Equation (4.2.1) above, one could conclude that (n+1)

h(R(d)

(n+1)

) ≤ h(V(d)

) + 3(N − 1) log(N + n + 1)

n+1 Y

(di + 1),

i=1 (n+1)

where V(d)

is the solution variety, N =

n X di + n n

i=0

(n+1)

is the dimension of P(H(d)

) and

(d) = (d0 , . . . , dn ) is a list of degrees. We believe that this bound is not sharp enough to understand the height of the multivariate resultant variety (whatever is the notion of global height h that had ever been chosen) and we then adress the question of finding sharper bounds for the height of the multivariate resultant variety. Here we try to find such a sharper bound by proving an Arithmetic Poisson Formula (Theorem 4.2.1 below). 4.2.1. Unitarily invariant height: An Arithmetic Poisson Formula. We begin by modifying slightly the norm of height of a polynomial when its variables have to be understood as the coefficients of other polynomials. This is the case of the resultant polynomial (n+1)

Res(d)

∈ Z[Aµ(i) : 0 ≤ i ≤ n, |µ| = di , µ ∈ Nn+1 ],

where (d) = (d0 , . . . , dn ) is a list of degrees of polynomials in n + 1 variables and the variables (i) Aµ are the generic coefficients of homogeneous polynomials X µ0 µn fi := A(i) µ X0 · · · Xn . |µ|=di (n+1)

As Res(d)

is a multi-homogeneous polynomial in each of the groups of variables Y (i) :=

(i)

{Aµ : |µ| = di , µ ∈ Nn+1 }, we define the unitarily invariant logarithmic Mahler’s measure of (n+1)

Res(d)

as (n+1)

mS(n+1) (Res(d)

) :=

1 νS [S(m) ]

Z S(n+1)

(n+1)

log |Res(d)

(f )|dνS (f ),


43

Q (n+1) (n+1) (n+1) where S(n+1) := ni=0 S(Hdi ), S(Hdi ) is the sphere of radius 1 in Hdi , with respect (n+1) to Bombieri-Weyl’s Hermitian norm, and dνS is the natural volume form in S endowed by the Bombieri-Weyl Riemannian structure on S(n+1) . Definition 4.2.1. We finally define the (unitarily invariant) height of the multi-variate re(n+1) sultant variety R(d) as: X (n+1) (n+1) hU (R(d) ) := mν (Res(d) ), ν∈Ω

where Ω = {∞} ∪ {p ∈ N : p is prime} is a complete class of representants of absolute values in Q, (n+1)

• m∞ (Res(d)

(n+1)

) := mS(n+1) (Res(d)

),

(n+1) mp (Res(d) )

(n+1)

• and := max{0, max{log |a|p : a is a coefficient of Res(d) p ∈ N prime, and | · |p : Q → R+ is the p-adic absolute value. (n+1)

As we may choose Res(d)

}} for every

with integer coefficients, this amounts to say (n+1)

hU (R(d)

(n+1)

) := mSn+1 (Res(d)

).

For technical reasons we may be interested in the following “density” quantity: (n+1)

(n+1) R(d)

:=

hU (R(d)

)

D(d)

,

Q where D(d) = ni=0 di is the corresponding Bézout number of the over-determined case. Then, the following statement is the technical statement of Main Theorem 1.3.1 of the Introduction. Theorem 4.2.1 (Arithmetic Poisson Formula). With the previous notations, for every degree list (d) = (d0 , . . . , dn ) ∈ Nn+1 of positive integers, the following equality holds: n 1 Y HR n (n) (n+1) R(d0 ) + J2 + Hn − , R(d) = di + n 2 d0 i=1 Q where (d0 ) := (d1 , . . . , dn ), D(d0 ) := ni=1 di , R = d0n+n and, as in previous sections, Hr denotes the r-th harmonic number, whereas J2 satisfies the following inequalities: n

n

j=1

j=1

X X πK(Nj , Lj )dj (dj + n) 1 1 − ≤ J2 = ≤− ≤ 0, 4 (Nj + 1)β(Nj − Lj + 1, Lj + 1)n 8(dj + n) (n+1)

(n+1)

where Nj + 1 = dim Hd0 , Lj + 1 = dim Hd0 the function introduced in Lemma 4.1.4.

as complex vector spaces and K(N, L) is

4.2.2. Proof of Theorem 4.2.1. As we already observed, there is a unique irreducible (and hence primitive) polynomial (d+1) Res(d)

∈ Z[

n [

{Aiµ : µ ∈ Nn+1 , |µ| = di }],

i=0 (n+1)

such that R(d)

(n+1)

= VP(H(n+1) ) (Res(d)

) (cf. Theorem 1.6.1 of [CD, 05]). We write

(d)

(n+1)

Res(d) (n+1)

to represent the value of Res(d) (n+1) Res(d)

(f0 , . . . , fn ), (n+1)

at the point f := (f0 , . . . , fn ) ∈ H(d)

.

This polynomial satisfies the Homogeneous Poisson Formula (cf. Proposition 1.6.2 of [CD, 05], for instance) which claims:

44

4. ARITHMETIC POISSON FORMULA AND APPLICATIONS (n+1)

With these notations, for every list f := (f0 , . . . , fn ) ∈ H(d) (n+1)

Res(d)

(n)

(f0 , . . . , fn ) = Res(d0 ) (f1,d1 , . . . , fn,dn )d0

, the following equality holds: Y a f0 (ζ)mζ ,

ζ∈VA (f1 ,...,fn )

where: • VA (f1 , . . . , fn ) ⊆ Cn is the variety of the common affine zeros of f1 , . . . , fn , • a fi (ζ) denotes the value of the affinization a fi at ζ ∈ Cn , namely a fi (ζ) := fi (1, ζ1 , . . . , ζn ), where ζ := (ζ1 , . . . , ζn ) ∈ Cn , • fi,di = fi (0, X1 , . . . , Xn ) is the equation fi viewed at the infinity hyper–plane {X0 = 0}, • mζ is the multiplicity of ζ as zero of (f1 , . . . , fn ) and, • (d0 ) = (d1 , . . . , dn ) ∈ Nn . (n)

Note that up to a subset of zero measure in H(d0 ) , we may always assume mζ = 1, for all ζ ∈ VA (f1 , . . . , fn ). As we already observed, (n+1)

(n+1) R(d)

hU (R(d)

:=

D(d)

(n+1)

) =

mS(n+1) (Res(d) D(d)

) ,

and this is the quantity we consider from now on. Let us denote by Rn+1 (f0 ) the following (n+1) integral for fixed f0 ∈ S(Hd0 ): Z (n+1) (n,n) Rn+1 (f0 ) := log |Res(d) (f0 , . . . , fn )|dS(d0 ) (f1 , . . . , fn ), (n,n+1)

S(d0 )

Q (n+1) (n,n+1) ) and (d0 ) = (d1 , . . . , dn ). Namely, a product of n spheres where S(d0 ) := ni=1 S(Hdi representing lists of coefficients of n polynomials in n + 1 variables of respective degrees Q (n,n) (n) d1 , . . . , dn . Along the proof we also consider S(d0 ) := ni=1 S(Hdi ) which are spheres of coefficients of n polynomials in n variables of respective degrees d1 , . . . , dn . Observe that Z 1 (n+1) (n+1) (4.2.2) R(d) = Rn+1 (f0 )dS(Hd0 )(f0 ). (n,n+1) (n+1) (n+1) D(d) vol[S(d0 ) × S(Hd0 )] S(Hd0 ) Next, we use the Homogeneous Poisson Formula to conclude Z (n) (n,n+1) Rn+1 (f0 ) = d0 log |Res(d0 ) (f1,d1 , . . . , fn,dn )|dS(d0 ) + (n,n+1) S(d0 ) Z (4.2.3) X (n,n+1) + log |f0 (ϕ−1 . 0 (ζ))|dS(d0 ) (n,n+1)

S(d0 )

ζ∈VP (f1 ,...,fn )

According to Lemma 4.1.5 above, we have (n) mS(n,n+1) (Res(d0 ) ) 0

(4.2.4)

(d )

(n) = J mS(n,n) (Res(d0 ) ) + J2 , (d0 )

where J=

n Y ì + 1 ≤ 1, mi + 1

−

j=1

i=1

D

0

n n X X D(d0 ) (mj − `j ) D(d0 ) (mj − `j ) ≤ J2 ≤ − ≤ 0, 4dj (`j + 1) 4dj (mj + 2)

(n)

j=1

where d(dj ) is the degree of Res(d0 ) in the group of variables given as the set of coefficients of the j−th equation fj,dj and, in our case, we have: di + n di + n − 1 (n+1) mi + 1 = dimC (Hdi )= , ì + 1 = . n n−1


45

After some calculations we have: J=

n Y i=1

and −

n di + n

≤ 1.

n n X X D(d0 ) D(d0 ) (mj − `j ) D(d0 ) =− =− , 4dj (`j + 1) 4n 4 j=1

−

j=1

n n n X X X D(d0 ) (mj − `j ) D(d0 ) (mj − `j ) D(d0 ) ≤− =− ≤ 0. 4dj (mj + 2) 8dj (mj + 1) 8(dj + n) j=1

j=1

j=1

Then, Equation (4.2.4) becomes: (n) mS(n,n+1) (Res(d0 ) ) 0

(4.2.5)

(d )

where J=

n Y i=1

n di + n

(n) = J mS(n,n) (Res(d0 ) ) + J2 , (d0 )

≤ 1, −

n X D(d0 ) D(d0 ) ≤ J2 ≤ − ≤ 0. 4 8(dj + n) j=1

With these notations, Equation (4.2.3) becomes the following one: Rn+1 (f0 ) = (4.2.6)

(n,n+1) d0 νS [S(d0 ) ]J

Z +

(n) mS(n,n) (Res(d0 ) ) 0

+ J2 +

(d )

(n,n+1)

X

log |f0 (ϕ−1 0 (ζ))|dS(d0 )

(n,n+1)

S(d0 ) (n)

.

ζ∈VP (f1 ,...,fn )

(n)

As mS(n,n) (Res(d0 ) ) = D(d0 ) R(d0 ) , we have proven (d0 )

(4.2.7)

(n,n+1)

Rn+1 (f0 ) = d0 D(d0 ) νS [S(d0 )

(n)

]J(R(d0 ) + J2 ) + J(f0 ),

where: J2 , D(d0 )

J2 := and Z J(f0 ) :=

(n,n+1)

X

(n,n+1) S(d0 )

log |f0 (ϕ−1 0 (ζ))|dS(d0 )

.

ζ∈VP (f1 ,...,fn )

Then, from Equation (4.2.2) we conclude: (4.2.8) Z 1 (n+1) (n+1) R(d) = Rn+1 (f0 )dS(Hd0 )(f0 ) = (n,n+1) (n+1) (n+1) D(d) νS [S(d0 ) × S(Hd0 )] S(Hd0 ) Z 1 (n+1) (n) J(f0 )dS(Hd0 )(f0 ) = = J R(d0 ) + J2 + (n,n+1) (n+1) (n+1) D(d) νS [S(d0 ) × S(Hd0 )] S(Hd0 ) 1 (n) ES(H (n+1) ) [J(f0 )], = J R(d0 ) + J2 + (n,n+1) d0 D(d) νS [S(d0 ) ] where ES(H (n+1) ) [J(f0 )] = d0

Z

1 (n+1) S(Hd0 )

(n+1) S(Hd ) 0

J2 satisfies: n

X 1 1 − ≤ J2 ≤ − ≤ 0. 4 8(dj + n) j=1

(n+1)

J(f0 )dS(Hd0

)(f0 ).

46


We may use the double fibration formula (Proposition 2.2.8 in Chapter 2 above) to conclude: (n,n+1)

J(f0 ) = D(d0 ) νS [S(d0 )

(4.2.9)

]EPn (C) [log |f0 ◦ ϕ−1 0 |],

where EPn (C) [log |f0 ◦

ϕ−1 0 |]

1 = νn [Pn (C)]

Z Pn (C)

log |f0 (ϕ−1 0 (ζ))|dPn (C)(ζ).

Namely, we have: (n,n+1)

ES(H (n+1) ) [J(f0 )] = D(d0 ) νS [S(d0 ) d0

]ES(H (n+1) ) [EPn (C) [log |f0 ◦ ϕ−1 0 |]] = d0

(n,n+1)

= D(d0 ) νS [S(d0 )

]E.

In particular, using Theorem 4.1.1 above we have: HR HR (n,n+1) 1 (n,n+1) d0 Hn − = D(d) νS [S(d0 ) ] Hn − . ES(H (n+1) ) [J(f0 )] = D(d0 ) νS [S(d0 ) ] 2 d0 2 d0 d0 Replacing this equality in the last row of Equation (4.2.8) we have: (n+1) R(d)

(4.2.10)

=J

(n) R(d0 )

+ J2

1 + 2

HR Hn − , d0

as wanted. The following Corollary is a more technical statement of Corollary 1.3.2 as stated at the Introduction. Corollary 4.2.2. With the same notations, we have: n 1 d0 n 1 n (n+1) Y (n) R(d0 ) ≤ log +O , R(d) − di + n 2 d0 + n n i=1

In particular, the straightforward inductive argument yields n 1X di n (n+1) R(d) ≤ + O(1), log 2 di + n i=0

and D(d) (n+1) hU (R(d) ) ≤ 2

n X

log

i=0

di n di + n

! +c ,

for some constant c > 0. Proof.– Simply observe that Hn = log(n) + O

1 , n

and d0 + n 1 d0 + n d0 1 HR = log +O ≥ log +O . d0 R d0 R Then, 1 HR 1 d0 + n 1 1 d0 n 1 Hn − ≤ log(n) − log +O = log +O . 2 d0 2 d0 n 2 d0 + n n Corollary 4.2.3. With the same notations as above, let γ∆ be the Gaussian distribution (n+1) on H(d) induced by Bombieri-Weyl’s norm (cf. Subsection 2.2.2 for more details on this probability distribution). Then, for every ε > 0, the probability (with respect to γ∆ ) that a

4.3. CHOW FORMS AND AFFINE ELIMINATION POLYNOMIALS (n+1)

sequence (f0 , . . . , fn ) ∈ H(d) bounded by:

(n+1)

satisfies that log|Res(d) n−1 X

47

(f0 , . . . , fn )| is greater than ε−1 is

! di n + c ε, log di + n i=0 Q for some positive constant c > 0, where D(d) = ni=0 di . D(d) 2

Proof.– First of all, note that logarithmic Mahler’s measure is,Qin fact, an expectation: The expected value of the multi–variate resultant on the product ni=0 S(Hdi ). Then, using Markov’s inequality we immediately have: ! n−1 h i D X d n (d) (n+1) i + c ε. Probf ∈Qni=0 S(Hd ) log|Res(d) (f0 , . . . , fn )| ≥ ε−1 ≤ log i 2 di + n i=0

Now, in order to transform into a statement about the Gaussian distribution, let us recall (n+1) that Res(d) is a multi-homogeneous polynomial and its degree is the group of variables Q corresponding to the coefficients of fi is j6=i dj . Thus, for every system f = (f0 , . . . , fn ) ∈ (n+1)

H(d)

, fi 6= 0 for all i, 0 ≤ i ≤ n, we have:

(n+1) log Res(d)

f0 fn ,..., ||f0 ||d0 ||fn ||dn

  n X Y = log|Res(n+1) (f0 , . . . , fn )| −  dj  log ||f ||di . (d) i=0

j6=i

According to the relations described in Subsection 2.2.2, we thus conclude: (n+1) Probγ∆ [log|Res(d) (f0 , . . . , fn )|

≥ε

−1

n Y X + ( di ) logkf kdi ] ≤ i=0 j6=i

D(d) ≤ 2 As kf k2∆ =

n−1 X i=0

log

di n di + n

! + c ε.

P kf k2di , we immediately conclude the inequality stated in the Corollary.

4.3. On the average Mahler’s measure of the Chow forms and affine elimination polynomials for zero-dimensional varieties We now return to some elementary Elimination aspects in the zero-dimensional complete in(n) tersection case. As above, let H(d) be the class of all systems f = (f1 , . . . , fn ) of homogeneous (n)

polynomial equations of respective degrees deg(fi ) = di , where (d) = (d1 , . . . , dn ). Let γ∆ (n) be the Gaussian distribution in H(d) induced by Bombieri-Weyl’s Hermitian product. There ˜ ⊆ H(n) of probability zero (with respect to γ (n) and also with respect to is a subvariety Σ (d) (n) H(d) )

∆

(n) H(d)

˜ the following two varieties are Lebesgue measure in such that for all f ∈ r Σ, zero-dimensional (i.e. they are just a finite set of points): VP (f ) := {x ∈ Pn (C) : fi (x) = 0, 1 ≤ i ≤ n}, VA (f ) := {x ∈ Cn : a fi (x) = 0, 1 ≤ i ≤ n}. (n) ˜ its cardinal equals the Bézout number D(d) = Qn di . The Moreover, for every f ∈ H(d) r Σ, i=1 (n) ˜ variety Σ is the case over the discriminant variety Σ ⊆ P(H ) (here notations follow these of (d)

(n) (n) ˜ and VP (f ), [BCSS, 98]). We simply say “f a generic system in H(d) ” to mean f ∈ H(d) r Σ, VA (f ) are respectively called projective and affine zero-dimensional complete intersection (n) varieties for a generically given system f ∈ H(d) . (n)

Two are the main elimination polynomials for a generically given f ∈ H(d) :

48


• The Chow form (also van der Waerden’s U-resultant): Let us introduce some new variables {U0 , . . . , Un } and the linear form U = U0 X0 + . . . + Un Xn . The Chow form (also U-resultant) of VP (f ) is the homogeneous polynomial: (n+1)


(f1 , . . . , fn , U) ∈ C[U0 , . . . , Un ].

(n) ˜ be • Elimination polynomial: With the same notations as above, let f ∈ H(d) r Σ given and let p ∈ C[X1 , . . . , Xn ] be and additional affine polynomial. We define the elimination polynomial of p with respect to VA (f ) as the characteristic polynomial χp,VA (f ) (T ) ∈ C[T ] of the following endomorphism

ξp : C[VA (f )] −→ C[VA (f )]. h 7−→ p·h Note that this characteristic polynomial χp satisfies: Y χp,VA (f ) (T ) := (T − p(ξ)), ξ∈VA (f ) (n)

where f is generically given in H(d) . We now wish to find estimates for the following two quantities: ECh := Ef ∈γ (n) [mS(H (n+1) ) (ChowVP (f ) )], 1

∆

and Eχp := Ef ∈γ (n) [mS 1 (χp,VA (a f ) )]. ∆

Namely, we wish to compute estimates for the logarithmic Mahler’s measure of Chow forms (n) and elimination polynomials for randomly chosen f ∈ H(d) with respect to the Gaussian (n)

distribution γ∆ . Moreover in many cases we can also be interested on the expected value (n+1) (1) of Eχp for a randomly chosen p ∈ Hd0 with respect to the Gaussian distribution γ∆ in (n+1)

Hd0

. Namely, we are also interested on bounding Eχ := Ep∈γ (1) [Eχa p ], ∆

where

ap

∈ C[X1 , . . . , Xn ] is the affinization of p as above.

Corollary 4.3.1. With the same notions and notations as above, we can conclude: Qn di ECh ≤ i=1 (log n + c), 2 and Q Qn di ( ni=1 di ) d0 Hn 1 ≤ i=0 Eχ ≤ log n + γ + O . 2 2 n Proof.– From the equalities discussed in Proposition 2.2.4, we immediately have Qn di ECh ≤ i=1 (log n + c). 2 As for the second inequality, we begin by recalling Jensen’s inequality from complex analysis (cf. [Ru, 87]) to conclude: Z X 1 log|χp,VP (f ) (z)|dνS (z) = max{0, log|p(ζ)|}, 2π S 1 ζ∈VA(f )

(n)

for every p ∈ C[X1 , . . . , Xn ] and for every generically given f ∈ H(d) . On the other hand, using Proposition 2.2.4 we have Eχp = mf ∈S(n+1) [mS 1 (χp,VA (f ) )]. (d)

4.3. CHOW FORMS AND AFFINE ELIMINATION POLYNOMIALS

49

Thus, putting together both claims, we conclude:   Z X 1  E χp = max{0, log|p(ζ)|} dνS (f ). (n+1) νS [S(d) ] f ∈S(n+1) (d) ζ∈VA (f ) Now, proceeding as in the Subsection 2.2.4 we also conclude: Eχp = D(d) EPn (C) [max{0, log|p ◦ ϕ−1 0 |}] for every p ∈ C[X1 , . . . , Xn ]. Now let us introduce the following quantity Z max{0, log|f (1, z)|} 1 dνS (f ). E(z) = (n+1) (n+1) (1 + kzk2 )n+1 νS [S(Hd0 )] S(Hd0 ) Combining Lemma 4.6 of [BP, 09a] with Fubini’s Theorem, we immediately conclude: Z D(d) Eχ = E(z)dz. νP [Pn (C)] Cn (n+1)

Now we concentrate our analysis on E(z). Let us introduce the following subset S(Hd0

)z ⊂

(n+1) S(Hd0 ): (n+1) S(Hd0 )z

f∈

:=

(n+1) S(Hd0 )

1 : |f (1, 0, . . . , 0)| ≥ (1 + kzk2 )d0 /2

.

(n+1)

Proceeding with the unitary invariance on S(Hd0 ) of Bombieri-Weyl’s metric (as in the proof of Theorem 4.1.1) we easily conclude: Z 1 max{0, d0 /2 · log|1 + kzk2 | + log|f (1, 0, . . . , 0)|} dνS (f ). E(z) := (n+1) (1 + kzk2 )n+1 ) νP [S(Hd0 )] S(Hdn+1 0 (n+1)

As f ∈ S(Hd0

)z if and only if 0≤

d0 log(1 + kzk2 ) + log|f (1, 0, . . . , 0)|, 2

we immediately conclude: Z

1

E(z) :=

(n+1)

νP [S(Hd0

)]

S(Hdn+1 )z

d0 /2 · log|1 + kzk2 | + log|f (1, 0, . . . , 0)| dνS (f ). (1 + kzk2 )n+1

0

Namely, we have E(z) ≤ E1 (z) + E2 (z), where E1 (z) := E2 (z) :=

Z

1 (n+1)

νS [S(Hd0 1

(n+1) )z 0

)]

S(Hd

Z

d0 log(1 + kzk2 ) dνS (f ), 2(1 + kzk2 )n+1 log|f (1, 0, . . . , 0)| dνS (f ). kzk2 )n+1

(n+1) (1 + )z νS [S(Hd0 )] S(Hd(n+1) 0 (n+1) ∈ S(Hd0 ), we have kf k2d0 =

Now, observe that as f 1, where k·kd0 is Bombieri-Weyl’s norm and |f (1, 0, . . . , 0)| is the absolute value of the coefficient of z0d0 in f . Then, according to the definition of Bombieri-Weyl’s norm (cf. Subsection 2.1.1), we have: |f (1, 0, . . . , 0)| ≤ kf k2d0 = 1. (n+1)

Thus, we conclude log|f (1, 0, . . . , 0)| ≤ 0 for all f ∈ S(Hd0 ) and E2 (z) ≤ 0. Hence, we conclude (n+1) d0 log(1 + kzk2 ) νS [S(Hd0 )z ] E(z) ≤ E1 (z) = . 2(1 + kzk2 )n+1 νS [S(H (n+1) )] d0

50


In particular, d0 log(1 + kzk2 ) . 2(1 + kzk2 )n+1 Finally, putting all these inequalities together we conclude: Z D(d) d0 log(1 + kzk2 ) Eχ ≤ dz. νP [Pn (C)] Cn (1 + kzk2 )n+1 Thus, proceeding as in the proof of Theorem 4.1.1, we conclude D(d) · d0 Eχ ≤ Hn , 2 and the statement follows. E(z) ≤

CHAPTER 5

Resumen en Castellano de los contenidos del Trabajo Fin de M´ aster 5.1. Introducci´ on Este manuscrito consiste en una serie de resultados altamente técnicos cuyo contexto solo puede ser definido como “matem´ aticas interdisciplinares”. La principal motivación de esta investigaci´ on es el Dise˜ no y An´ alisis de Algoritmos Numéricos Eficientes en Geometr´ıa Algebraica. Dicha motivaci´ on viene al menos de tres marcos matemáticos habitualmente distantes: Complejidad Computacional, Análisis Numérico y Geometr´ıa Algebraica. La mayor´ıa de estos resultados se centran en cuestiones relacionadas con el Problema 17 de Smale. Este problema fue enunciado en la famosa lista de 18 problemas para el próximo siglo escrita por S. Smale tras una propuesta de V.I. Arnold (cf. [Sm, 00]). Problema 5.1.1 (Problema 17 de Smale). Can a zero of n complex polynomial equations in n unknowns be found approximately, on the average, in polynomial time with a uniform algorithm? Este problema fue resuelto en [BP, 09a] (cf. también [BP, 11a]) utilizando un algoritmo aleatorio Las Vegas que resuelve ecuaciones polinomiales multivariadas complejas en tiempo promedio cuadr´ atico en el tama˜ no de la entrada (O(N 2 )). A´ un existen atractivas preguntas abiertas que, por una parte, pueden ayudar a mejorar aspectos técnicos de la soluci´ on propuesta y, por otra parte, continuar las investigaciones motivadas por esta solución. Este manuscrito pretende trabajar en ambas direcciones. Una de las principales caracter´ısticas de la solución conocida al Problema 17 de Smale ya estaba en la base del trabajo desarrollado en colaboración por M. Shub y S. Smale entre 1981 y 1995 (que culmin´ o en la famosa serie de manuscritos conocidos como “Bézout series”, que van de [SS, 93] a [SS, 94]). Los autores no fueron capaces de dar un algoritmo eficiente que resolviese ecuaciones polinomiales multivariadas (y esto motivó que Smale enunciase su Problema 17). Sin embargo, se˜ nalaron algunos de los indicios para encontrar dicho algoritmo. Debemos se˜ nalar las siguientes: (1) Usar el n´ umero de condicionamiento no lineal µnorm para acotar la complejidad de algoritmos basados en deformaciones homotópicas. (2) Usar técnicas de Geometr´ıa Integral para enfrentar el estudio de propiedades en media de cantidades relacionadas con la complejidad. De hecho, la soluci´ on del Problema 17 de Smale en [BP, 09a] y [BP, 11a] está basada en la introducci´ on de nuevas técnicas de Geometr´ıa Integral relacionando n´ umeros de condicionamiento y variedades algebraicas. Por ejemplo, un ingrediente fundamental fue el estudio del valor medio de µ2norm sobre las rectas (reales) esféricas en la esfera definida por el producto herm´ıtico de Bonbieri-Weyl. Este manuscrito también sigue este ámbito de investigación estudiando las esperanzas de otras cantidades relacionadas con variedades algebraicas. Por ejemplo, estudiamos esperanzas de algunas propiedades métricas de las variedades algebraicas. De hecho en la Subsecci´ on 5.2.5 se muestra la distancia promedio entre dos variedades proyectivas de intersecci´ on completa aleatoriamente escogidas. La principal motivación de esta cuesti´ on es la versi´ on numérica del Nullstellensatz enunciado como el Problema 5.2.2 siguiente. Notar que, por ejemplo, el Teorema 5.2.1 demuestra que la esperanza de la distancia promedio entre dos variedades algebraicas proyectivas de intersecci´ on completa aleatoriamente escogidas coincide con el di´ ametro observable de Pn (C) como en [Gr, 99] y sus referencias. También estudiamos la esperanza de la separación de los ceros de un variedad de dimensión cero e 51

52

´ 5. RESUMEN EN CASTELLANO DE LOS CONTENIDOS DEL TRABAJO FIN DE MASTER

intersecci´ on completa elegida aleatoriamente. La separación de las soluciones es un tema clásico en el tratamiento numérico-simbólico de ecuaciones polinomiales y sus soluciones, y también est´ a motivado por la presencia de redundancias en el output de un algoritmo de resolución no universal (cf. el enunciado del Problema 5.2.3 y [BP, 06] o [CaGiHeMaPa, 03] para algoritmos de resoluci´ on no universales de ecuaciones polinomiales). Nuestros principales resultados sobre las cotas de la separación son enunciadas en la Subsección 5.2.5 y las demostraciones aparecen a lo largo del manuscrito. El principal resultado de cotas de la separaci´ on promedio es el Teorema 5.2.3 que muestra que la esperanza promedio de la separaci´ on de una variedad de dimensi´ on cero e intersecci´ on completa, elegida aleatoriamente, es polinomial en el tama˜ no del input, al contrario que en las cotas conocidas del caso peor, que son doblemente exponenciales en el n´ umero de variables. Como el lector puede observar, demostrar estos resultados implica necesariamente involucrar aspectos de Geometr´ıa Algebraica, Análisis Numérico, Geometr´ıa Integral, Teor´ıa de la Medida, Probabilidad, Teor´ıa de la Intersección y otras que muestran la naturaleza interdisciplinar de este manuscrito. A pesar de la belleza y potencial aplicabilidad de nuestros resultados sobre la distancia y separaci´ on promedio, estos no son el principal logro de este manuscrito. El principal resultado de estas p´ aginas es la F´ ormula Aritm´ etica de Poisson, enunciado como el Teorema Principal 5.2.1 siguiente (Subsección 5.2.2). Este resultado principal a˜ nade dos nuevos marcos de trabajo a la naturaleza interdisciplinar de estas páginas: Geometr´ıa Diofántica y Teor´ıa de la Eliminación. De la Geometr´ıa Diof´ antica tomamos la noción de altura de los polinomios, los puntos proyectivos con coordenadas en cuerpos de n´ umeros y las variedades algebraicas diofánticas. Esta noción inicialmente fue desarrollada por A. Weil para medir el tama˜ no de un punto proyectivo x ∈ Pn (k), donde k es un cuerpo de n´ umeros. Inspirados por la Desigualdad de Bézout, muchos autores usaron la noci´ on de altura de una variedad diof´ antica como la noción principal para probar una Desigualdad Aritmética de Bézout (cf. [Ph, 86], [Ph, 91], [Ph, 95], (n+1) [BGS, 94]). Aqu´ı estudiamos la altura de la variedad resultante multivariada R(d) (cf. Corolario 5.2.2). (n+1) La variedad resultante R(d) es el objeto de estudio central de la Teor´ıa de la Eliminaci´ on. Es una hipersuperficie proyectiva algebraica inmersa en el espacio proyectivo del espacio dado por los coeficientes de sistemas dobredeterminados de ecuaciones polinomiales homogéneas (f0 , . . . , fn ), con grados determinados por una lista (d) = (d0 , . . . , dn ). Esta variedad resul(n+1) tante multivariada R(d) es el conjunto de los ceros comunes de un polinomio irreducible mul(n+1)

tihomogéneo Res(d)

con coeficientes enteros y, por tanto, también es un objeto diofántico.

(n+1) Res(d)

Al polinomio se le conoce como resultante multivariada y su principal caracter´ıstica viene dada por la siguiente propiedad: Dados f0 , . . . , fn ∈ C[x0 , . . . , xn ], n + 1 polinomios homogéneos. Sea (d) = (d0 , . . . , dn ) la lista de grados. Entonces, el sistema de ecuaciones sobredeterminado f = (f0 , . . . , fn ) tiene (n+1) un cero com´ un proyectivo si y solo si Res(d) (f0 , . . . , fn ) = 0. Es decir, (n+1)

∃ζ ∈ Pn (C), f0 (ζ) = · · · = fn (ζ) = 0 ⇔ Res(d)

(f0 , . . . , fn ) = 0.

Nótese que la parte izquierda de esta equivalencia es una fórmula de primer orden que contiene, como instancias particulares, todos los problemas NP-completos conocidos. Es una versión proyectiva del Nullstellensatz. En otras palabras, el conocimiento y la computaci´ on (n+1) de Res(d) es un ingrediente fundamental para entender muchos problemas intratables de Complejidad Computacional. Aqu´ı somos modestos, pues solo intentamos encontrar cotas finas en la altura logar´ıtmica de la resultante multivariada. Con este fin, podemos primero aplicar la Desigualdad Aritmética de Bézout. Sin embargo, este intento no proporciona cotas superiores finas (cf. la Ecuaci´ on

5.2. ENUNCIADO DE LOS PRINCIPALES RESULTADOS

53

(5.2.2)). Por tanto debemos proceder con otra tecnolog´ıa matemática. Las resultantes multivariadas satisfacen una famosa f´ ormula clásica conocida como F´ ormula de Poisson. Esta (n+1) (n) fórmula relaciona Res(d) con Res(d0 ) , donde (d) = (d0 , . . . , dn ), (d0 ) = (d1 , . . . , dn ). Esta (n+1)

relación es, por ejemplo, u ´til para computar el grado de Res(d) , el cual se sabe que es igual n Y X que dj . El lector podr´ a encontrar un enunciado preciso de la Fórmula de Poisson en i=0 j6=i

[CD, 05] y sus referencias. Combinando la F´ ormula de Poisson y nociones estándar de altura no se obtiene una Fórmula Aritmética de Poisson (como ocurre con las cotas del grado). La razón es que la noci´ on usual de altura (como la de Philippon en [Ph, 91]) solo lleva a desigualdades y, por tanto, (n+1) (n) no ayuda a tener una igualdad que exprese la relación entre h(Res(d) ) y h(Res(d0 ) ). Como principal contribuci´ on de esta memoria, hemos logrado una F´ ormula Aritmética de Poisson al reemplazar la noci´ on usual de altura por una variación basada en el producto herm´ıtico (n+1) de Bombieri-Weyl. Llamamos a esta altura la altura unitariamente invariante hU (Res(d) ), y se adapta mucho mejor para entender las relaciones existentes entre la altura de los ceros y la altura de las ecuaciones. La F´ ormula Aritmética de Poisson aparece como el Teorema Principal 5.2.1 siguiente y su demostración se desarrolla a lo largo del trabajo. Las nociones precisas y notaciones requeridas para entender el Teorema Principal 5.2.1 están descritas en la Subsecci´ on 5.2.2. La noci´ on de altura unitariamente invariante es descrita en la Subsecci´ on 5.2.2 y discutida con mayor detalle en el manuscrito. A partir de nuestras cotas de la altura unitariamente invariante de la resultante multivariada podemos concluir casi inmediatamente cotas finas de la medida de Mahler de dos objetos centrales de la Teor´ıa de la Eliminación: los polinomios de Chow (también conocidos como U-resultante de van der Waerden) y los polinomios de Eliminaci´ on. Estas cotas se presentan en el Corolario 5.2.3 y estas dos clases de polinomios son discutidas en la Subsección 5.2.3. Las demostraciones aparecen con detalle a lo largo del manuscrito. Con el fin de probar la F´ ormula Aritmética de Poisson también demostramos un resultado técnico que, debido a la belleza de su enunciado, hemos llamado “armon´ıa” de la esperanza de la media de Mahler. Esto es el Teorema Principal 5.2.4 siguiente, cuya prueba aparece también a lo largo del manuscrito. Este enunciado simplemente demuestra una igualdad entre la esperanza de la medida de Mahler de un polinomio aleatorio y una diferencia de dos n´ umeros arm´ onicos. Este resultado nos da esperanza de poder encontrar una estimación fina (o una igualdad precisa) para la esperanza de la función zeta de Akazuka de un polinomio aleatorio, pero de momento es una conjetura. 5.2. Enunciado de los principales resultados 5.2.1. Algunas nociones b´ asicas y notaciones. Sean n, m ∈ N dos enteros positivos. (n+1) Para toda cota del grado d, denotamos por Hd el espacio vectorial complejo de todos los (n+1) polinomios homogéneos en C[X0 , . . . , Xn ] de grado d. Con las mismas notaciones, Pd es el espacio vectorial complejo de todos los polinomios de grado como mucho den C[X1 , . . . , Xn ]. Ambos espacios vectoriales son obviamente isomorfos de dimensión d+n y el isomorfismo n (n+1)

(n+1)

viene dado por la aplicaci´ on a : Hd −→ Pd , que asocia a cada polinomio homogéneo (n+1) (n+1) a f ∈ Hd su “afinizaci´ on” f := f (1, X1 , . . . , Xn ) ∈ Pd . Escribimos simplemente Hd y Pd (omitiendo el super-´ındice (n+1) ) cuando no puede haber confusión con el n´ umero de variables. Fijemos el n´ umero de variables n + 1 a lo largo de esta sección. Para toda lista de grados Q (m) positivos (d) = (d1 , . . . , dm ) ∈ Nm , denotamos por H(d) := m i=1 Hdi al espacio vectorial complejo de todas las listas de m polinomios f = (f1 , . . . , fm ), donde fi ∈ C[X0 , . . . , Xn ] es un polinomio homogéneo de grado di para todo i, 1 ≤ i ≤ m. Análogamente denotamos Q (m) por P(d) := m i=1 Pdi al conjunto de listas de m polinomios afines g = (g1 , . . . , gm ), donde gi ∈ C[X0 , . . . , Xn ] es un polinomio af´ın de grado como mucho di . La afinización obviamente

54

´ 5. RESUMEN EN CASTELLANO DE LOS CONTENIDOS DEL TRABAJO FIN DE MASTER (m)

(m)

define un isomorfismo entre P(d) y H(d) . Por tanto, la siguiente aplicación es un isomorfismo de espacios vectoriales complejos: (m)

(m)

H(d) −→ P(d) . (f ) = (f1 , . . . , fm ) 7−→ (a f ) := (a f1 , . . . ,a fm ) La dimensi´ on compleja de ambos espacios vectoriales cumple: m X di + n (m) (m) dimC (H(d) ) = dimC (P(d) ) := . n i=1

En ocasiones consideramos el espacio proyectivo complejo definido por uno de estos espacios. (m) Denotamos por P(H(d) ) este espacio vectorial complejo y denotamos por N(d) la dimensi´ on compleja de este espacio proyectivo. Cuando no haya ninguna confusión simplemente denotamos por N esta dimensi´ on. (m) Para toda lista f = (f1 , . . . , fm ) ∈ H(d) , sea VP (f ) ⊆ Pn (C) la variedad proyectiva compleja de sus ceros comunes: VP (f ) = {X ∈ Pn (C) : fi (X) = 0, 1 ≤ i ≤ m} ⊆ Pn (C), (m)

Similarmente, para toda lista g = (g1 , . . . , gm ) ∈ P(d) , debemos considerar la variedad af´ın algebraica VA (g) ⊆ Cn de sus ceros comunes: VA (g) = {X ∈ Cn : gi (X) = 0, 1 ≤ i ≤ m} ⊆ Cn . Sea ϕ0 la inmersi´ on est´ andar de Cn en Pn (C), (5.2.1)

ϕ0 :

Cn −→ Pn (C) \ {X0 = 0}. (X1 , . . . , Xn ) 7−→ (1 : X1 : . . . : Xn )

Obsérvese que VA (a f ) se puede identificar con VP (f )∩(Pn (C)r{X0 = 0}). Entonces, tenemos (m) VA (a f ) = ϕ−1 0 (VP (f )) para todo f ∈ H(d) . En lo que sigue, denotaremos Q por d := max{di : 1 ≤ i ≤ m} al máximo de los grados de umero de Bézout asociado a la lista de la lista (d), y por D(d) := m i=1 di denotaremos al n´ grados (d). (m) Como en [SS, 93] o [BCSS, 98] (Sec. 12.1) podemos equipar H(d) con el producto herm´ıtico unitariamente invariante de Bombieri-Weyl, (m)

(m)

h·, ·i∆ : H(d) × H(d) −→ C. p Denotamos por k·k∆ := h·, ·i∆ a la norma inducida por este producto herm´ıtico y denotamos (m) (m) por S(H(d) ) a la esfera de radio uno centrada en el origen en H(d) , definida por este producto herm´ıtico. Es decir, (m) (m) S(H(d) ) := {f ∈ H(d) : kf k2∆ = 1}. En lo que sigue utilizaremos la serie armónica y los n´ umeros armónicos para expresar nuestros enunciados. Pasemos a recordar algunos aspectos básicos sobre n´ umeros armónicos. Denotamos por Hr al r-ésimo n´ umero armónico. Es decir, Hr :=

r X 1 . k k=1

Recuérdese que el n-ésimo n´ umero armónico Hn cumple: lim (Hn − log(n)) = γ,

n7→∞

donde γ es el n´ umero de Euler-Mascheroni. Recordamos también que ∞ X ζ(m, n + 1) (Hn − log(n)) = γ + , m m=2


55

donde ζ(m, n + 1) es la funci´ on zeta de Hurwitz. Además, también tenemos (Hn − log(n)) = γ +

1 1 1 − + − ε, 2 2n 12n 120n4

para 0 0. Una segunda propiedad de esta F´ ormula Aritmética de Poisson es que proporciona cotas superiores para la probabilidad (con la distribución gaussiana) de que la resultante multivariada crezca demasiado r´ apido. Esto aparece en el manuscrito con más detalle. 5.2.3. Esperanza de la medida de Mahler de las Formas de Chow y de los Polinomios de Eliminaci´ on. Una tercera aplicación de la Desigualdad Aritmética de Poisson proporciona cotas superiores finas de la esperanza de la medida de Mahler de las formas de Chow y de los polinomios de Eliminación con respecto a variedades algebraicas de dimensi´ on cero e intersecci´ on completa. Ahora volvemos a algunos aspectos elementales de Teor´ıa de la Eliminación en el caso de (n) dimensión cero e intersecci´ on completa. Como antes, sea H(d) la clase de todos los sistemas de ecuaciones polinomiales homogéneas de grados dados por la lista (d) = (d1 , . . . , dn ). Sea (n) (n) γ∆ la distribuci´ on gaussiana en H(d) inducida por el producto herm´ıtico de Bombieri-Weyl. ˜ ⊆ H(n) de probabilidad cero (con respecto a γ (n) y también con Existe una subvariedad Σ ∆

(d)

(n) H(d) )

(n) H(d)

˜ las variedades respecto a la medida de Lebesgue en tal que para todo f ∈ r Σ, a n VP (f ) ⊆ Pn (C) y VA ( f ) ⊆ C son variedades cero dimensionales sin puntos singulares (i.e. un conjunto finito de puntos no singulares). (n) ˜ el cardinal satisface: Más a´ un, para todo f ∈ H(d) r Σ ](VP (f )) = ](VA (a f )) = D(d) =

n Y

di .

i=1

˜ es el cono sobre la variedad discriminante Σ ⊆ P(H(n) ). Cuando decimos que La variedad Σ (d) (n) (n) ˜ “f es un sistema genérico en H ” queremos decir que f ∈ H r Σ, y debemos reclamar que (d)

(d)

las variedades VP (f ), VA (f ) sean (proyectiva y af´ın, respectivamente) variedades de dimensi´ on (n) cero e intersecci´ on completa para un sistema genéricamente dado f ∈ H(d) . (n)

Los principales polinomios de eliminación para un f ∈ H(d) genéricamente dado son dos: • La forma de Chow (también U-resultante de van der Waerden’s): Introducimos algunas nuevas variables {U0 , . . . , Un } y la forma lineal U = U0 X0 + . . . + Un Xn . La forma de Chow (también U-resultante) de VP (f ) es el polinomio homogéneo: (n+1)


(f1 , . . . , fn , U) ∈ C[U0 , . . . , Un ].


59 (n)

• Polinomio de Eliminaci´ on: Con las mismas anteriores notaciones, sea f ∈ H(d) r ˜ Σ dado y sea p ∈ C[X1 , . . . , Xn ] y polinomio af´ın adicional. Definimos el polinomio de eliminaci´ on de p con respecto a VA (a f ) como el polinomio caracter´ıstico χp,VA (a f ) (T ) ∈ C[T ] dado por la siguiente identidad: Y (T − p(ξ)). χp,VA (a f ) (T ) := ξ∈VA (a f )

Ahora queremos encontrar estimaciones para las dos cantidades siguientes: ECh := Ef ∈γ (n) [mS(H (n+1) ) (ChowVP (f) )], 1

∆

y Eχp := Ef ∈γ (n) [mS 1 (χp,VA (a f ) )], ∆

(n) γ∆

(n)

donde es la distribuci´ on gaussiana en H(d) inducida por el producto herm´ıtico y la norma de Bombieri-Weyl. A saber, queremos computar estimaciones para la medida logar´ıtmica de Mahler de las formas (n) de Chow y los polinomios de eliminación de f ∈ H(d) aleatoriamente escogidos. Más a´ un, en muchos casos también podemos estar interesados en la esperanza del valor de Eχp para un (n+1)

(1)

(n+1)

p ∈ Hd0 aleatoriamente escogido con respecto a la distribución gaussiana γ∆ en Hd0 También estamos interesados en acotar

.

Eχ := Ep∈γ (1) [Eχa p ], ∆

donde

ap

∈ C[X1 , . . . , Xn ] es la afinización de p como antes.

Corolario 5.2.3. Con las mismas nociones y notaciones anteriores podemos concluir que: Qn di ECh ≤ i=1 (log n + c), 2 y Q Qn ( ni=1 di ) d0 Hn di 1 Eχ ≤ ≤ i=0 log n + γ + O . 2 2 n 5.2.4. Cierta “armon´ıa” en la esperanza de la medida de Mahler. La medida de Mahler ha sido usada como una medida efectiva del tama˜ no de los polinomios. Se la ha considerado m´ as apropiada que la norma canónica k·k2 a causa de su buen comportamiento con respecto al producto de polinomios (es decir, M (f ·g) = M (f )·M (g)). La norma de BombieriWeyl es la u ńica competidora seria, puesto que satisface la desigualdad de Bombieri (aparece detallada en el manuscrito). Sin embargo, la medida de Mahler se mantiene como la principal medida arquimediana de polinomios con coeficientes en un cuerpo de n´ umeros y, por tanto, la medida de Mahler es la principal cantidad a estudiar en Geometr´ıa Aritmética/Diofántica. El logaritmo de la medida de Mahler es, de hecho, unaQesperanza del logaritmo del polinomio a lo largo del producto de circunferencias complejas ni=1 S 1 ⊆ Cn . Esta es una propiedad que la medida de Mahler comparte con la norma de Bombieri-Weyl. En los primeros noventa, P. Philippon (cf. [Ph, 91], [Ph, 95]) introdujo algunas variaciones de la medida logar´ıtmica de Mahler, reemplazando el producto de circunferencias por el producto de esferas e incluso la esfera S 2n−1 ⊆ Cn . Su principal resultado en [Ph, 91] fue un teorema que aparece enunciado con detalle en el manuscrito y que simplemente compara la esperanza del valor de la medida logar´ıtmica de Mahler cuando escogemos o bien el producto Qn 1 o bien la esfera S 2n−1 . S i=1 En este manuscrito probamos el comportamiento armónico de la esperanza de la medida (n+1) logar´ıtmica de Mahler de un polinomio aleatoriamente escogido f ∈ Hd . Es decir, (n+1)

(n+1)

Teorema Principal 5.2.4. Sea S(Hd ) la esfera unidad en Hd con respecto al producto herm´ıtico de Bombieri-Weyl y sea ϕ0 : Cn → Pn (C) el encaje can´ onico del espacio af´ın

60


(n+1) en su clausura proyectiva. Sea R := d+n la dimensi´ on compleja de Hd . Definamos la n esperanza: E := Ef ∈S(H (n+1) ) [EPn (C) [log |a f ◦ ϕ−1 0 |]]. d

Tenemos: (5.2.3)

d 0≤E= 2

HR 1 Hn − ≤ d(log(n) + γ) + O . d n

5.2.5. Esperanza de la distancia y de la separaci´ on. Como un producto colateral de nuestros estudios técnicos también conseguimos mejoras en el estudio de varios aspectos relacionando distancia proyectiva (Fubini-Study) y variedades algebraicas de intersecci´ on completa. Asumimos que Pn (C) est´ a dotado con la estructura métrica de Fubini-Study y dR : Pn (C) × Pn (C) → R+ denota la distancia riemanniana. La distancia riemanniana dR : Pn (C) × Pn (C) → R+ es la distancia definida por la estructura riemanniana de Pn (C). Es decir, sea π : Cn+1 r {0} → Pn (C) la proyección canónica en el espacio complejo proyectivo. Dados x, y ∈ Pn (C) y x1 , y1 ∈ Cn+1 r {0} tales que π(x1 ) = x, π(y1 ) = y, la distancia riemanniana (también llamada Fubini-Study) dR (x, y) está definida por |hx1 , y1 i| , dR (x, y) = arccos kx1 k2 ky1 k2 donde h·, ·, i : Cn+1 × Cn+1 → C es el producto herm´ıtico canónico en Cn+1 , y kx1 k2 , ky1 k2 son las normas respectivas con respecto a este producto herm´ıtico. Como en [BCSS, 98], también introducimos la distancia “proyectiva” entre dos puntos proyectivos x, y ∈ Pn (C) mediante la siguiente identidad: dP (x, y) := sin dR (x, y). Utilizando esta distancia proyectiva estudiamos varias cantidades diferentes relacionando distancia entre ceros y variedades algebraicas. A lo largo del manuscrito discutimos dos tipos principales de preguntas. La primera pregunta est´ a basada en el siguiente Problema clásico: (m)

(s)

Problema 5.2.1 (Nullstellensatz de Hilbert). Sean f ∈ H(d) y g ∈ H(d0 ) dos sistemas de ecuaciones polinomiales homogéneas. Asumamos que las variedades proyectivas VP (f ) y VP (g) son de intersecci´ on completa sin puntos singulares y con respectivas co-dimensiones m y s. Decidir si VP (f ) ∩ VP (g) es vac´ıa o no. Nótese que la resultante multivariada fue definida para responder a preguntas como este Nullstellensatz de Hilbert. De hecho, asumamos m = n y s = 1 en el enunciado del Problema (n) (n+1) 5.2.1 anterior. Ahora, dados f = (f1 , . . . , fn ) ∈ H(d) y g ∈ Hd0 , el Nullstellensatz de Hilbert tiene una respuesta negativa si y solo si (n+1)

Res(d0 ) (g, f1 , . . . , fn ) = 0, donde (d0 ) = (d0 , d1 , . . . , dn ) y (d) = (d1 , . . . , dn ). Desde el an´ alisis numérico tenemos una aproximación diferente a este Problema. (m)

(s)

Problema 5.2.2 (Versi´ on Numérica del Nullstellensatz). Sean f ∈ H(d) y g ∈ H(d) dos sistemas de ecuaciones polinomiales homogéneas. Sea z ∈ Pn (C) un cero aproximado de f con cero asociado ζ ∈ VP (f ). Decidir si ζ ∈ VP (g) o no. Recordamos que un cero aproximado z ∈ Pn (C) de f con cero asociado ζ ∈ VP (f ) es un punto tal que 1 dP (Nfk (z), VP (f )) ≤ dP (Nfk (z), ζ) ≤ 2k−1 , 2 donde Nfk (z) ∈ Pn (C) es la k-ésima iteración del operador de Newton proyectivo aplicado a z. Entonces, si VP (f ) ∩ VP (g) = ∅, las cotas inferiores para la distancia m´ınima entre VP (f ) y VP (g) podr´ıan ser utilizadas para controlar el n´ umero de iteraciones requeridas antes de


61

llegar “suficientemente lejos” de VP (g) (cf. [BCSS, 98], [BP, 09b], [BP, 11b] para discusiones sobre estas ideas y [KP, 96], [HMPS, 00] o [KPS, 01] para otros acercamientos al Nullstellensatz). Estudiamos dos cantidades principales: Z 1 dP (x, y)dνn−m (x)dνn−s (y), δav [f, g] := νn−m [VP (f )]νn−s [VP (g)] VP (f )×VP (g) dP (VP (f ), VP (g)) := min{dP (x, y) : x ∈ VP (f ), y ∈ VP (g)}, donde dνn−m y dνn−s son la medida canónica de Haussdorff de respectivas dimensiones n − m y n−s (cf. [BP, 07] para m´ as detalles) y νn−m [VP (f )], νn−s [VP (g)] son los vol´ umenes de VP (f ) and VP (g) respectivamente con respecto a estas medidas de Haussdorff. La cantidad δav [f, g] es la esperanza de la distancia de dos puntos puntos aleatoriamente escogidos x ∈ VP (f ) and y ∈ VP (g). Como VP (f ) y VP (g) son ambos conjuntos compactos, dP (VP (f ), VP (g)) es un m´ınimo. N´ otese que si m + s ≤ n, y las variedades VP (f ) y VP (g) son de co-dimensiones m y s respectivamente, entonces VP (f ) ∩ VP (g) 6= ∅ y dP (VP (f ), VP (g)). Por tanto, los u ńicos casos de estudio interesantes son aquellos en los que m + s ≥ n + 1. En este caso, VP (f ) ∩ VP (g) = ∅ para f y g aleatoriamente escogidos. Probamos los dos resultados siguientes: Teorema 5.2.1. Con estas notaciones, tenemos: EP(H(m) )×P(H(s) ) [δav (f, g)] = (d0 )

(d)

(m)

1 1− 2n + 1

,

(s)

(m)

(s)

donde P(H(d) )×P(H(d0 ) ) es el producto de dos distribuciones can´ onicas en P(H(d) ) y P(H(d0 ) ) inducido por su estructura riemanniana. Nótese que esta igualdad se cumple independientemente de los valores de m y s e independientemente de el hecho de que VP (f ) ∩ VP (g) sea vac´ıo o no. Este resultado también es sorprendente puesto que la esperanza de la distancia que obtenemos coincide con el diámetro observable de del espacio proyectivo complejo en los términos de Gromov (cf. [Gr, 99], cap´ıtulo 3 1/2, y sus referencias). Teorema 5.2.2. Con las mismas notaciones, asumiendo que m = n y s ≥ 1, la siguiente (s) desigualdad se cumple para todo g ∈ H(d0 ) fijado tal que VP (g) es no singular de co-dimensi´ on s y VP (f ) ∩ VP (g) = ∅: (5.2.4)

Ef ∈P(H(n) ) [dP (VP (f ), VP (g))] ≥ (d)

2s − 1 D(d) + 2 deg(VP (g))D(d)

, e 2s s

donde e es la base Q del logaritmo natural, deg(VP (g)) es el grado de la variedad proyectiva VP (g) y D(d) = si=1 di es el n´ umero de Bézout asociado a la lista (d) = (d1 , . . . , dn ). En particular, tenemos EP(H(m) )×P(H(s) ) [dP (VP (f ), VP (g))] ≥ (d0 )

(d)

D(d)

2s − 1 Q 1 + 2 si=1

d0i e2 s2

,

donde (d0 ) = (d01 , . . . , d0s ) es la lista de grados asociada a la segunda clase de ecuaciones. Como corolario inmediato concluimos: Corolario 5.2.5. Con las mismas notaciones anteriores, tenemos: (1) Para s = 1, se cumple la siguiente desigualdad: E(f,g)∈P(H(n) )×P(H(n+1) ) [dP (VP (f ), VP (g))] ≥ (d)

(d0 )

1 . D(d) (1 + 2e2 d0 )

(2) Para s ≥ 3, se cumple la siguiente desigualdad: E(f,g)∈P(H(n) )×P(H(s) ) [dP (VP (f ), VP (g))] ≥ (d)

(d0 )

5 . D(d) (1 + 2D(d0 ) )

62


Otra materia cl´ asica en la resoluci´ on de ecuaciones polinomiales es el estudio de cotas inferiores para la separaci´ on de los ceros de un sistema de ecuaciones. Esto es relevante tanto en métodos basados en b´ usqueda binaria (métodos del tipo divide y vencerás) como en aquellos basados en deformaciones homot´ opicas de tipo no universal (i.e. aquellos métodos que no siguen todas las soluciones a lo largo del camino prescrito como los de la conjetura de ShubSmale en [SS, 94], o aquellos basados en conjuntos questor como en [BP, 09a] y [BP, 11a]). Una cuesti´ on de este tipo es la siguiente: (n)

Problema 5.2.3 (Redundancia en Resolución Numérica No Universal). Sea f ∈ H(d) un sistema dado de n ecuaciones polinomiales multivariadas homogéneas. Suponer que también se nos dan dos puntos z1 , z2 ∈ Pn (C) tales que ambos son ceros aproximados de f con ceros asociados ζ1 , ζ2 ∈ VP (f ) respectivamente. Decidir si ζ1 = ζ2 o no. Como ambos z1 y z2 son ceros aproximados de f , obviamente tenemos dP (ζ1 , ζ2 ) ≤ dP (Nfk (z1 ), ζ1 ) + dP (Nfk (z2 ), ζ2 ) + dP (Nfk (z1 ), Nfk (z2 )) ≤ ≤

2

+ dP (Nfk (z1 ), Nfk (z2 )).

22k−1

Si ζ1 6= ζ2 , hay una cantidad sep(f ) (la separación de VP (f )) tal que sep(f ) ≤ dP (ζ1 , ζ2 ) y, por tanto, concluimos que sep(f ) −

2 22k−1

≤ dP (Nfk (z1 ), Nfk (z2 )).

En particular, las cotas inferiores para sep(f ) son esenciales para determinar el n´ umero k de iteraciones del método de Newton requeridas para decidir si ζ1 6= ζ2 . Una de las cotas inferiores para la separación más famosa es la cota DMM, introducida para el caso de sistemas univariados por J. Davenport (cf. [Da, 88]), quien lo atribuye a ideas de Kurt Mahler y M. Mignotte (cf. [Mg, 95]). Estas cotas son extendibles al caso multivariado en varias direcciones. Sugerimos el manuscrito [EMT, 10] para resultados recientes de estas estimaciones y sus referencias. Para las generalizaciones queremos remarcar [De, 97]. Otro ejemplo de c´ omo extender el contraejemplo de Mignotte al caso af´ın multivariado puede ser encontrado en [CHMP, 01]. Todos estos resultados pretenden demostrar una cota inferior para la separaci´ on que sea n doblemente exponencial en el n´ umero de variables (i.e. Ω(22 ) como cota inferior). Como esta cota inferior se obtiene para el peor caso, queremos explorar su valor medio. Solamente nos restringimos al caso proyectivo, aunque el caso af´ın es un tema para ser tratado en futuros trabajos. También nos restringimos al caso cero dimensional. (m)

´ n 5.2.1 (Cotas para la separación). Sea f = (f1 , . . . , fn ) ∈ H(d) una secuencia de Definicio n ecuaciones polinomiales homogéneas y asumamos que la variedad proyectiva VP (f ) ⊆ Pn (C) es cero dimensional (i.e. un conjunto finito de puntos). (1) Definimos la separaci´ on media de los ceros de f mediante la siguiente igualdad: X 1 dP (ζ, ζ 0 ). sepav (f ) := ]VP (f )(]VP (f ) − 1) ζ, ζ 0 ∈ VP (f ) ζ 6= ζ 0 (2) También consideramos la cota cota para la separaci´ on como es habitual: sep(f ) := min{dP (ζ, ζ 0 ) : ζ, ζ 0 ∈ VP (f ), ζ 6= ζ 0 }. Demostramos los siguientes resultados: Teorema 5.2.3. Con estas notaciones, la siguiente igualdad se cumple: √ √ s 6(3 − 7) 1 Ef ∈P(H(n) ) [sepav (f )] ≥ , 3 8 d (N + 1/2)n (d)


63

donde d = max{di : 1 ≤ i ≤ n}, (d) = (d1 , . . . , dn ) y N = N(d) es la dimensi´ on del espacio (n)

proyectivo complejo P(H(d) ). Nótese que esta cota inferior no es doblemente exponencial en el n´ umero de variables, es un polinomio en d, n y en el tama˜ no de la entrada N . Como para la cota para la separaci´ on, también probamos: Teorema 5.2.4. Con estas notaciones, también tenemos: √ 3− 7 (N + 1/2)−1/2 , Ef ∈P(H(n) ) [sep(f )] ≥ 4eD(d) d3/2 (d) Q umero de Bézout. donde d, n y N son como en el Teorema anterior y D(d) = ni=1 di es el n´ Nótese que, en esta ocasi´ on, la cota inferior es simplemente exponencial en el n´ umero de variables y no doblemente exponencial como en el peor caso de la cota DMM.

Bibliography [BP, 06]

C. Beltr´ an, L.M. Pardo, On the Complexity of Non Universal Polynomial Equation Solving: Old and New Results, in “Foundations of Computational Mathematics, Santander 2005”, L.M. Pardo, A. Pinkus, E. S¨ ulli & M. Todd eds., London Mathematical Society Lecture Notes Series 331, Cambridge University Press, 2006, 1–35. [BP, 07] , Estimates on the Distribution of the Condition Number of Singular Matrices, Foundations of Computational Mathematics 7 (2007), 87–134. [BP, 08] , On Smale’s 17th Problem: A Probabilistic Positive Solution, Foundations of Computational Mathematics 8 (2008), 1–43. [BP, 09a] , Smale’s 17th Problem: Average Polynomial Time to Compute Affine and Projective Solutions, J. of the Amer. Math Soc. 22 (2009), 363–385. [BP, 09b] , Efficient polynomial system-solving by numerical methods, J. Fixed Point Theory and Applications 6 (2009) 65–85. [BP, 11a] , Fast linear homotopy to find approximate zeros of polynomial systems, Found. of Comput. Math. 11 (2011), 95–129. , Efficient Polynomial System Solving by Numerical Methods, “Randomization, Relax[BP, 11b] ation, and Complexity in Polynomial Equation Solving”, L. Gurvits, P. Pebay, J.M. Rojas, D. Thompson, eds., Contemporary Mathematics, vol. 556, American Mathematical Society, 2011, 1–35. [BoP, 08] C.E. Borges, L.M. Pardo, On the probability distribution of data at points in real complete intersection varieties, J. of Complexity 24 (2008), 492–523. [BCSS, 98] L. Blum, F. Cucker, M. Shub, and S. Smale, Complexity and real computation, Springer-Verlag, New York, 1998. [BGS, 94] J.B. Bost, H. Gillet, C. Soulé, Heights of projective varieties and positive Green forms. J. Amer. Math. Soc. 7 (1994), 903–1027. [CHMP, 01] D. Castro, K. H¨ agele, J.E. Morais, L.M. Pardo, Kronecker’s and Newton’s approaches to solving: a first comparison, J. of Complexity 17 (2001), 212–303. [CaGiHeMaPa, 03] D. Castro, M. Giusti, J. Heintz, G. Matera, L.M. Pardo, The Hardness of Polynomial Equation Solving, Foundations of Computational Mathematics 3 (2003), 347–420. [CD, 05] E. Cattani, A. Dickenstein, Introduction to residues and resultants. In A.Dickenstein, I.Z.Emiris (Eds.): Solving Polynomial Equations: Foundations, Algorithms, and Applications, Algorithms and Comput. Math., 14, Springer-Verlag, 2005, 1–61. [Ch, 99] K.-K. Choi, On the Distribution of Points in Projective Space of Bounded Height, Transactions of the Amer. Math. Soc. 352 (1999), 1071–1111. [DAKrSo, 12] C. D’Andrea, T. Krick, M. Sombra, Heights of Varieties in Multiprojective spaces and Arithmetic Nullstellens¨ atze, Manuscript, 2012. [Da, 88] J.H. Davenport, Cylindrical algebraic decomposition, Technical Report 88-10, School of Mathematical Sciences, University of Bath, 1988. [De, 97] J.P. Dedieu, Estimations for the Separation Number of a Polynomial System, J. Symbolic Computation 24 (1997), 683–693. [De, 06] J.P. Dedieu, Points fixes, zéros et la méthode de Newton, Springer, 2006. [EMT, 10] T.Z. Emiris, B. Mourrain, E.P. Tsigaridas, The DMM bound: Multivariate (aggregate) separation bounds, arXiv: 1005.5610, 2010. [Fe, 69] H. Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. [Fu, 84] W. Fulton, Intersection Theory, Ergeb. Math. Genzeb. 2, Springer, Berlin, 1984. [Ga, 59] W. Gautschi, Some elementary inequalities relating to the gamma and incomplete gamma function, J. Math. Phys. 38 (1959), 77–81. [Gr, 99] M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Birkh¨ auser Boston, 1999. [HMPS, 00] K. H¨ agele, J.E. Morais, L.M. Pardo, M. Sombra, The intrinsic complexity of the Arithmetic Nullstellensatz, Journal of Pure and Applied Algebra 146 (2000), 103–183. [HW, 60] G.H. Hardy, E.M. Wright, An introduction to the theory of numbers, Oxford: Clarendon Press (1960).

65

66

[He, 83] [Ho, 94] [Kr, 83] [KP, 96] [KPS, 01] [Le, 94] [MK, 01] [Mg, 89] [Mg, 95] [Ph, 86] [Ph, 91] [Ph, 94] [Ph, 95] [GQ, 08] [Re, 01a] [Re, 01b] [Ru, 87] [SS, 93] [SS, 93b]

[SS, 96] [SS, 94]

[Sm, 00] [Vo, 84]

BIBLIOGRAPHY

J. Heintz, Definability and Fast Quantifier Elimination in Algebraically Closed Fields, Theoret. Comput. Sci. 24 (1983), 239–277. R. Howard, Analysis on homogeneous spaces, Class notes spring 1994. Royal institute of technology, Stocholm.nn D. Kershaw, Some extensions of W. Gautschi’s inequalities for the gamma function, Math. Comp. 41 (1983), 607–611. T. Krick, L.M. Pardo, A computational method for diophantine approximation, Algorithms in Algebraic Geometry, Progress in Mathematics 143, L. T. Krick, L.M. Pardo, M. Sombra, Sharp estimates for the arithmetic Nullstellensatz, Duke Mathematical Journal 109 (2001), 521–98. P. Lelong, Mesure de Mahler et calcul des constantes universelles pour les polynˆ omes de n variables, Math. Ann. 299 (1994), 673–695. D. McKinnon, An arithmetic analogue of Bézout’s Theorem, Compositio. Math. 126 (2001), 147–155. M. Mignotte, Mathématiques pour le Calcul Formel, Presses Universitaires de France, 1989. M. Mignotte, On the Distance Between Roots of a Polynomial, Appl. Alg. Eng. Commun. Comput. 6 (1995), 327–332. ´ P. Philippon, Critères pour l’independance algébrique, Inst. Hautes Etudes Sci. Publ. Math. 64 (1986), 5–52. P. Philippon, Sur des hauteurs alternatives, I, Math. Ann. 289 (1991), 255–283. P. Philippon, Sur des hauteurs alternatives, II, Ann. Inst. Fourier (Grenoble) 44 (4) (1994), 1043–1065. P. Philippon, Sur des hauteurs alternatives, III, J. Math. Pures appl. 74 (1995), 345–365. F. Qi, B.-N. Guo, Wendel’s and Gautschi’s inequalities: Refinements, extensions, and a class of logarithmically completely monotonic functions, Appl. Math. and Comput. 205 (2008), 281–290. G. Rémond, Elimination multi-homogène, Chapter 5 of “Introduction to algebraic independence theory”, Lecture Notes in Math. 1752, Springer, 2001, 53–81. G. Rémond, Géométrie diophantienne multiprojective, Chapter 7 of “Introduction to algebraic independence theory”, Lecture Notes in Math. 1752, Springer, 2001, 95-131. W. Rudin, Real and Complex Analysis, McGraw Hill Book Co., New York, 1987. M. Shub, S. Smale, Complexity of Bézout’s Theorem I: Geometric Aspects, J. of the Amer. Math. Soc. 6 (1993), 450–501. , Complexity of Bezout’s theorem. II. Volumes and probabilities, Computational algebraic geometry (Nice, 1992), Progr. Math., vol. 109, Birkh¨ auser Boston, Boston, MA, 1993, pp. 267– 285. , Complexity of Bezout’s theorem. IV. Probability of success; extensions, SIAM J. Numer. Anal. 33 (1996), no. 1, 128–148. , Complexity of Bezout’s theorem. V. Polynomial time, Theoret. Comput. Sci. 133 (1994), no. 1, 141–164, Selected papers of the Workshop on Continuous Algorithms and Complexity (Barcelona, 1993). S. Smale, Mathematical problems for the next century, Mathematics: Frontiers and Perspectives, Providence, RI: Amer. Math. Soc., 2000, pp. 271–294. W. Vogel, Lectures on Results on Bézout’s Theorem, Lecture Notes, Tata Institute of Fundamental Research No 74, Bombay, 1984.