Hâbases for polynomial interpolation and system solving

Advances in Computational Mathematics 0 (2003) ?–?

1

H–bases for polynomial interpolation and system solving H. Michael Möller, a Thomas Sauer b a

FB Mathematik der Universit¨ at Dortmund, D-44221 Dortmund, Germany E-mail: [email protected] b Mathematisches Institut, Universit¨ at Erlangen–N¨ urnberg, Bismarckstr 1 21 , D–91054 Erlangen, Germany E-mail: [email protected]

The H–basis concept allows similarly to the Gr¨ obner basis concept a reformulation of nonlinear problems in terms of linear algebra. We exhibit parallels of the two concepts, show properties of H–bases, discuss their construction and uniqueness questions, and prove that n polynomials in n variables are under mild conditions already H–bases. We apply H–bases to the solving of polynomial systems by the Eigenmethod and to multivariate interpolation. Keywords: Ideal bases, Gr¨ obner bases, multivariate polynomials, interpolation, systems of polynomial equations. AMS Subject classification: 65D05,65H10,13P10.

1.

Introduction

At a very early time, when even the notion of ideals was not commonly accepted, Macaulay introduced H–bases [14]. These special bases of polynomial ideals are also helpful in various branches of Numerical Analysis. In many applications, one is interested in getting a basis or a generating set for the linear vector space I ∩ Pd , where Pd is the space of all polynomials in x1 , . . . , xn of degree at most d and where I is an ideal like the ideal of all polynomials vanishing at a given set of points. Having an H–basis {f1 , . . . , fs } for the ideal I, then the set of all pi · fi with pi ∈ Pd−deg(fi ) , i = 1, . . . , s, linearly generates the finite dimensional vector space I∩Pd . Hence, restricting the pi ’s to a basis of Pd−deg(fi ) and canceling linearly dependent polynomials, one gets a basis for the linear space I ∩ Pd . Thus, H–bases can be seen as a tool for transforming

2

H. M. M¨ oller and T. Sauer / H–bases for interpolation and system solving

a problem in the infinite dimensional vector space of all polynomials in x1 , . . . , xn into a problem in one (or a series of) finite dimensional vector space(s) I ∩ Pd . A similar concept is that of Gröbner bases, see [1,3,7,11]. For these bases, one first has to order all power products (terms) xi11 · · · xinn linearly. Then, analogously to an H–basis, a Gröbner basis allows to find a linear generating system for the vector space I ∩ Ft , where the linear space Ft is generated by all terms less or equal to the term t = xi11 · · · xinn . If the terms are ordered primarily by degrees, then every I ∩ Pd is a space I ∩ Ft for a suitable term t. Hence, the Gröbner basis concept can be considered as “finer” than the H–basis concept. It has been studied thoroughly over the last 20 years and several applications to Numerical Analysis are known, see for instance [18,22]. However, the rigid ordering of terms in the Gr¨ obner basis access also has drawbacks. If the given ideal is eventually invariant under a group of permutations of x1 , . . . , xn , then the same does not hold for the Gr¨ obner basis. In other words, the “finer” structure of Gröbner bases has to be paid with the destruction of symmetries or invariances. Since the additional information on the spaces I ∩ Ft is not needed in many applications which only request bases for spaces I ∩ Pd , H–bases seem to be better suited. This paper is mainly addressed more to readers with background in Numerical Analysis than to those who are more familiar with Commutative Algebra. Therefore, we first explain at an elementary level the underlying common structure of H–bases and Gröbner bases and show parallels between the two concepts. The dimensions of the vector spaces I ∩ Pd are described using the Hilbert function in section 3. There we also present modules of syzygies, a tool for analyzing linear dependencies among polynomials, and their dimensions. In section 4 we summarize the known methods for computing H–bases and turn then, in section 5, to regular sequences. There we show in thm. 5.3, that n polynomials in n variables are already an H–basis, if their homogeneous parts of maximal degree have no other zero in common than 0. In contrast to Gr¨ obner bases, which are under mild additional conditions uniquely determined by the given ideal and the term ordering, H–bases are not unique. We give in section 6 an exact description of the structure of H–bases for a given ideal. The last two sections are devoted to applications. First we show in section 7 that Stetter’s Eigenmethod also works with H–bases in place of Gröbner bases and then finally in section 8 how H–bases can be used to solve multivariate polynomial interpolation problems.


2.

3

H–bases and Gr¨ obner bases

Here and in the following sections we consider polynomials in n variables x1 , . . . , xn with coefficients from a field K, say the field Q of rational numbers or the field R of real numbers. For short, we write P := K[x1 , . . . , xn ] . Given a set F ⊂ P, the set I :=

 X

 

hf f | hf ∈ P, only finitely many hf 6= 0





f ∈F

is the ideal generated by F. We express this dependence shortly by hFi := I. Note that the Hilbert basis theorem implies that F can always be replaced by a finite subset of F, hence we can always assume F to be finite. Finite sets generating an ideal are usually called (ideal) bases. In this section we want to introduce H–bases and some of their properties. This concept is very similar to the concept of Gr¨ obner bases. Therefore, we will briefly explain the underlying common structure following the exposition of Möller and Mora in [19]. Let Γ denote an ordered monoid, i.e. an abelian semigroup under an operation +, equipped with a total ordering ≺ such that, for all α, β, γ ∈ Γ, α≺β

=⇒

γ+α≺γ+β .

(2.1)

A direct sum P :=

M

Pγ(Γ)

γ∈Γ

is called a grading (induced by Γ) or briefly a Γ-grading if for all α, β ∈ Γ (Γ)

f ∈ Pα(Γ) , g ∈ Pβ

=⇒

(Γ)

f · g ∈ Pα+β .

Since the decomposition above is a direct sum, each polynomial f 6= 0 has a unique representation f=

s X

fγi ,

0 6= fγi ∈ Pγ(Γ) . i

i=1

Assuming that γ1 ≺ γ2 ≺ · · · ≺ γs , the Γ–homogeneous term fγs is called the maximal part of f , denoted by M (Γ) (f ) := fγs , and f − M (Γ) (f ) is called the reductum of f .

4


There are two major examples of gradings. The first one is the grading by degrees, (Γ)

Pd

:= {p ∈ P | p homogeneous of degree d}

∀d ∈ N0 .

(2.2)

Here, Γ = N0 with the natural total ordering. This grading is called the H-grading because of the homogeneous polynomials. Therefore we also write H in place of this Γ. The space of all polynomials of degree at most d can now be written as Pd :=

d M

(H)

Pk

.

(2.3)

k=0

The maximal part of a polynomial f 6= 0 is its homogeneous form of highest degree, M (H) (f ). For simplicity, let M (H) (0) := 0. Definition 2.1. A finite set G := {g1 , . . . , gs } ⊂ P \ {0} is called an H–basis of the ideal I := hg1 , . . . , gs i if, for all 0 6= p ∈ I, ∃h1 , . . . , hs : p =

s X

hi gi

deg(hi ) + deg(gi ) ≤ deg(p),

and

i = 1, . . . , s .

i=1

(2.4) The representation for p in (2.4) is also called its H–representation with respect to G. To obtain more insight into H–bases, we will give some equivalent definitions. First we need a more technical notion. Definition 2.2. For given f, f1 , . . . , fm ∈ P we say that f reduces to f˜ modulo F := {f1 , . . . , fm } if f˜ = f −

m X

gi fi ,

deg(f˜) < d,

i=1

holds with polynomials gi satisfying deg(gi ) ≤ deg(f ) − deg(fi ), i = 1, . . . , m. In this case we write f By

∗ −→

F

−→ f˜

F

.

we denote the transitive closure of the binary relation

f reduces modulo F to g if f

∗ −→

F

g.

−→

F

. We also say


5

Theorem 2.3. Let F := {f1 , . . . , fs } and I := hFi. Then the following conditions are equivalent i) F is an H–basis of I. ii) hM (H) (f1 ), . . . , M (H) (fs )i = h{M (H) (f ) | f ∈ I}i. iii) Every f ∈ I reduces modulo F to 0. Proof. i) =⇒ ii) follows by M (H) (f ) =

X

M (H) (hj )M (H) (fj ),

J := {j | deg(hj fj ) = deg(f ) }

j∈J

for arbitrary f ∈ I with H-representation f = hi fi . ii) =⇒ iii) follows by an inductive argument from P

0 6= f ∈ I =⇒ f if M (H) (f ) = iii) =⇒ i) Let

P

j∈J

F

f−

X

M (H) (hj )fj ∈ I

j∈J

M (H) (hj )M (H) (fj ). g0 := f

and M (H) (gi−1 ) =

−→ f˜ =

P

−→ g −→ 1

F

F

...

−→ g d

F

=0

M (H) (hij )M (H) (fj ), i = 1, . . . , d. Then f=

s X d X

M (H) (hij )fj .

j=1 i=1

Note that Hilbert’s Basissatz says that any ideal in P has a finite basis, hence, by ii), any ideal in P has an H–basis. The second major example of gradings leads to the Gröbner basis concept. Here, Γ := Nn0 with componentwise addition and equipped with a total ordering ≺ satisfying (2.1) and in addition 0 γ ∀γ ∈ Γ. (In the Gr¨ obner context, this ordering is called then admissible.) For arbitrary γ = (γ1 , . . . , γn ) ∈ Γ, the space (Γ) Pγ is then a vector space of dimension 1, namely Pγ(Γ) = {c · xγ11 · · · xγnn | c ∈ K} . The maximal part of a polynomial is now a product of a leading coefficient and a leading power product, M (Γ) (f ) = lc(f ) · lt(f ), lc(f ) ∈ K, lt(f ) a power product. The reduction f −→ f˜ is defined if there exists a polynomial g ∈ G such that lt(g) G

6

H. M. M¨ oller and T. Sauer / H–bases for interpolation and system solving ∗

(Γ) (f ) divides lt(f ) and then we set f˜ := f − M g. The relation −G→ is constructed M (Γ) (g) as above. A Gröbner basis G (with respect to a given admissible term order and a given ideal I) is a finite set of polynomials, say {g1 , . . . , gs }, generating the ideal I and satisfying one of the following equivalent conditions

i) Every f ∈ I has a representation f = si=1 hi gi , max lt(hi ) lt(gi ) ≤ lt(f ). ii) hM (Γ) (g1 ), . . . , M (Γ) (gs )i = h{M (Γ) (f ) | f ∈ I}i. iii) Every f ∈ I reduces modulo G to 0. P

The proof of this equivalence and many other equivalent conditions can be found in [19, Theorem 3.2] . If an admissible ordering is compatible with the semi-ordering by degrees, 0

deg(xγ ) < deg(xγ ) =⇒ γ ≺ γ 0 ,

γ, γ 0 ∈ Nd0 ,

then any Gr¨ obner-representation as given in i) is an H-representation, in other words, a Gröbner basis with respect to a degree compatible ordering is an H–basis as well. The converse is false as the following example shows. Example 2.4. Let f1 := x4 + 2x2 y 2 + y 4 − 1 and f2 := xy(x2 − y 2 ). Then these polynomials constitute already an H–basis. ( This will be an easy consequence of theorem 5.3.) If we order the power products first by degree and same degrees by lower powers in x first, then we get a degree compatible ordering called “ total degree ordering” in MAPLE. Since then h{M (Γ) (f ) | f ∈ I}i = hx3 y, x4 , x2 y 3 , xy 5 , y 7 i , every Gr¨ obner basis G w.r.t. this ordering contains at least 5 elements, for instance MAPLE computes G = {g1 , . . . , g5 } with g1 :=

x3 y − xy 3

= f2 ,

g2 := x4 + 2x2 y 2 + y 4 − 1 = f1 , g3 :=

3x2 y 3 + y 5 − y,

g4 :=

4xy 5 − xy,

g5 := 4y 7 − 4y 3 + 3x2 y. Obviously, this G-basis is an H–basis as well.


3.

7

Linear vector spaces and H–bases

The definition of H–bases shows that having an H–basis g1 , . . . , gs of an ideal I, the finite dimensional vector space of all p ∈ I with degree at most d, i.e. the space I ∩ Pd , is generated by all power product multiples xi11 · · · xinn · gj with i1 + . . . + in + deg(gj ) ≤ d. Hence, problems like the membership problem p ∈ I can be displayed as a problem in a finite dimensional vector space and solved by linear algebra techniques if an H–basis of I is known. In this section we consider some finite dimensional vector spaces in connection with ideals. Set, for given polynomials f1 , . . . , fs and d ∈ N0 , Vd (f1 , . . . , fs ) :=

( s X

hi M

(H)

(fi ) | hi ∈

(H) Pd−deg(fi ) ,

)

i = 1, . . . , s

,

(3.1)

i=1

(H)

where, as a convention, Pk = {0} if k < 0. Clearly, Vd (f1 , . . . , fs ) is a linear (H) subspace of Pd . For convenience, we also define for ideals I Vd (I) := {M (H) (f ) | f ∈ I, deg(f ) = d or f = 0} . (H)

This is a linear subspace of Pd as well. Fix an inner product (·, ·) in P, for instance the scalar product of the coefficient vectors or an inner product (f, g) := L(f g), where L is a strictly positive functional, e.g. an integral. Then, as shown by Sauer [23], the orthogonal com(H) plements of Vd (f1 , . . . , fm ) and Vd (I) in Pd are useful tools. To fix notations, let (H)

Pd

= Vd (f1 , . . . , fs ) ⊕ Wd (f1 , . . . , fs ),

(H) Pd

= Vd (I) ⊕ Wd (I) . (H)

We will also use the shorthand notation , for example Wd (I) = Pd to express this relationship.

Vd (I),

Definition 3.1. For given polynomials g1 , . . . , gm ∈ P let Syz (g1 , . . . , gm ) :=

(

(h1 , . . . , hm ) ∈ P

m

|

m X

hi gi = 0

)

i=1

be the module of syzygies with respect to g1 , . . . , gm . For every d ∈ N0 let Sd (g1 , . . . , gm ) :=

(

(h1 , . . . , hm ) ∈

O

(H) Pd−deg(gi )

|

m X i=1

hi M

(H)

(gi ) = 0

)

.

8


Lemma 3.2. Let F := {f1 , . . . , fs } ⊂ P be given. Then the following four statements hold. i) M (H) (F) =

M

Vd (f1 , . . . , fs ).

d∈N0

ii) Syz (M (H) (f1 ), . . . , M (H) (fs )) =

M

Sd (f1 , . . . , fs ).

d∈N0 (H)

iii) dim Vd (f1 , . . . , fs ) = dim Pd iv) dim Sd (f1 , . . . , fs ) =

s X

− dim Wd (f1 , . . . , fs ), ∀d ∈ N0 . (H)

dim Pd−deg(fi ) − dim Vd (f1 , . . . , fs ), ∀d ∈ N0 .

i=1

Proof. By homogeneity of the basis elements M (H) (fi ), each representation h = (H) (f ) splits into homogeneous representations for the homogeneous i i=1 gi M components of h. Hence i) follows. Similarly, looking at the homogeneous parts P (H) (f ) = 0 one gets ii). Since V (f , . . . , f ) = of a syzygy relation m i s d 1 i=1 hi M (H) Pd Wd (f1 , . . . , fs ), iii) follows immediately. Every syzygy in Sd (f1 , . . . , fs ) is a linear dependence relation among the degree d polynomials hi M (H) (fi ), hi a power product of degree d − deg(fi ) for i = 1, . . . , s. dim Sd (f1 , . . . , fs ) is hence the maximal number of linearly indepenP (H) dent relations of this kind. Since the si=1 dim Pd−deg(fi ) homogeneous degree d Ps

polynomials hi M (H) (fi ) generate Vd (f1 , . . . , fs ), the assertion follows.

The dimension of the finite dimensional spaces considered in lemma 3.2 can easily be described using the Hilbert function. This function is studied carefully for instance by Gröbner in [12] and [13]. For the distinction of Hilbert functions and affine Hilbert functions see also [9]. Definition 3.3. Let I ⊆ P be an ideal. The mapping HI : N0 −→ N0 ,

HI (d) := dim(Wd (I)),

is called the (homogeneous) Hilbert function of I. We call a

HI (d) :=

d X

HI (k),

aH

I

with

d ∈ N0 ,

k=0

the affine Hilbert function of I. For every ideal I ⊆ P there exists a constant T = T (I) such that HI (t) becomes a polynomial in t for t ≥ T . This is the so called Hilbert polynomial,


9

see [9]. Then, as the definition of a HI shows, also a HI (t) becomes a polynomial in t, the affine Hilbert polynomial. Its degree is one more than the degree of the (non-affine) Hilbert polynomial. Lemma 3.4. Let F := {f1 , . . . , fs } be an H–basis of the ideal I. Then the following four statements hold for all d ∈ N0 . i) dim Vd (I) = d+n−1 − HI (d). n−1 Ps i) ii) dim Sd (f1 , . . . , fs ) = HI (d) − d+n−1 + i=1 d+n−1−deg(f . n−1 n−1 a H (d). iii) dim(I ∩ Pd ) = d+n − I n Ps iv) dim(Syz (M (H) (f1 ), . . . , M (H) (fs ))) = a HI (d) − d+n + i=1 n

Here we declare the binomial

a b

d+n−deg(fi ) . n

to be 0 for all a with −a ∈ N.

Proof. There is the well known formula (H) dim(Pd )

=

!

d+n−1 , n−1 (H)

which is sometimes also called Hurwitz formula. Using Pd = Vd (I) ⊕ Wd (I) the definition of HI (d) gives i). Then theorem 2.3 ii) and lemma 3.2 iv) give ii). (H) Using the decomposition (2.3) one obtains Pk = Pk−1 ⊕ Pk . Hence dim(I ∩ Pd ) = dim(I ∩ Pd−1 ) + dim VI (d) = . . . =

d X

dim Vk (I) .

k=0

Hence iii) follows by i) and the definition of a HI (d). In the same way iv) can be deduced from iii) and lemma 3.2 ii).

4.

On the construction of H–bases

Macaulay introduced H–bases in [14, p. 39], using the intimate relation between polynomials in x1 , . . . , xn of degree at most d and homogeneous polynomials of degree d in x0 , . . . , xn based on the concept of the homogeneization of a polynomial. This relation is for instance described in the book of Cox, Little, and O’Shea [9]. But for characterizing H–bases, one also has to consider the connection of ideals generated by polynomials of P and of ideals generated by their homogenized polynomials. This was done by Macaulay in [14]. Details can be found for instance in the books of Gröbner [12,13].

10


Introducing an additional variable x0 , every polynomial p ∈ P with HP decomposition p = di=0 pi can be homogenized to Φd (p) :=

d X

pi ·

i=0

xd−i 0

=

xd0

x1 xn ·p ,..., x0 x0

.

Φd (p) is homogeneous of degree d and Φd (p)|x0 =1 = p. Hence Φd is a bijection between Pd and the space Hd of all homogeneous polynomials of degree d in x0 , . . . , xn . Obviously Φk (p) = xk−d 0 Φd (p) if k ≥ d = deg(p). If F is homogeneous (in x0 , . . . , xn ) and F |x0 =1 = f , then F = Φk (f ) for a k ≥ deg(f ). Let I be an ideal in P. Then denote by Φ(I) the ideal of n + 1-variate polynomials generated by all Φdeg(p) (p), p ∈ I. This ideal is a so–called homogeneous ideal (for short: H-ideal), since it is generated by homogeneous polynomials. Following Macaulay, we call Φ(I) the H-ideal equivalent to I, because of the bijection between I ∩ Pd and Φ(I) ∩ Hd for all d ∈ N0 . Since Φ(I) is an H-ideal, a polynomial F belongs to Φ(I) iff all its homogeneous components belong to Φ(I). If {F1 , . . . , Fs } is a basis of Φ(I), then every homogeneous Φd (f ) has a representation as a sum of homogeneous degree d polynomials, i.e., Φd (f ) =

s X

H i Fi ,

Hi ∈ Hd−deg(Fi ) , i = 1, . . . , s.

i=1

Since one may assume that no Fi is a multiple of x0 , the dehomogenization x0 = 1 gives f=

s X

hi fi with hi ∈ Pd−deg(fi )

i=1

where Fi = Φdeg(fi ) (fi ). This means that {f1 , . . . , fs } is an H–basis of I. This was the observation leading Macaulay to the introduction of the notion of H–bases. More precisely, the connection is as follows. Theorem 4.1. Let I ⊆ P be an ideal, let fi ∈ I and Fi := Φdeg(fi ) (fi ) for i = 1, . . . , s. Then {f1 , . . . , fs } is an H–basis of I iff {F1 , . . . , Fs } is a basis of the H-ideal equivalent to I. In [14, p. 40] Macaulay gave an example of an H–basis construction. There he claimed that “ the method given is a general one”. His method consists in a


11

succession of homogenizations and dehomogenizations of ideals and uses heavily that he knows a basis for the module of syzygies Syz (M (H) (f1 ), . . . , M (H) (fk )) for every intermediate ideal hf1 , . . . , fk i, k = 1, . . . , s. Unfortunately, he did not describe a general method how he constructed such bases. In the case of Gr¨ obner bases, there is the well–known Buchberger algorithm which can be interpreted as an algorithm for computing Gröbner bases by means of syzygies. (This is described for instance in the book of Adams and Loustaunau [1, p. 118ff], where syzygies and their connection to Gröbner bases are studied in detail.) The analogue for constructing H–bases by means of syzygies is based on the following result. Theorem 4.2 [23]. Let F := {f1 , . . . , fm } ⊂ P be a basis of I. Then the following conditions are equivalent. i) F is an H–basis. ii) If (g1 , . . . , gm ) ∈ Sd (M (H) (f1 ), . . . , M (H) (fm )) then p :=

m X

∗

gi fi −→ 0 . F i=1

Proof. The conclusion i) ⇒ ii) is trivial by Theorem 2.3 iii), since p ∈ I by construction. Hence, it remains to show ii) ⇒ i), where we follow an argumentation from [17]. Given f ∈ I, the goal is to use ii) to construct polynomials h1 , . . . , hm such that f=

m X

hj fj ,

deg (hj fj ) ≤ deg(f ).

(4.1)

j=1

Since f ∈ I, it can be written as f = If, in this representation,

Pm

j=1 gj fj

for some polynomials g1 , . . . , gm .

deg(f ) = d := max {deg (gj fj ) | j = 1, . . . , m} , then we already have the desired representation (4.1). If, on the other hand, this is not the case, then a cancellation of leading parts occured and we can apply

12


our assumption ii) to reduce the degree of the representation. For this purpose we define the homogeneous polynomials g˜j =

(

M (H) (gj ) 0

if deg (gj fj ) = d, if deg (gj fj ) < d,

j = 1, . . . , m,

and observe that (˜ g1 , . . . , g˜m ) ∈ Sd (M (H) (f1 ), . . . , M (H) (fm )) holds. Therefore, as in the proof iii)=⇒ i) of theorem 2.3, polynomials h1 , . . . , hm can be found such that m X

g˜j pj =

j=1

m X

hj pj ,

deg (hj pj ) < d.

j=1

Consequently, f=

m X

j=1

gj pj =

m X

(gj − g˜j ) pj +

j=1

m X

hj pj =

j=0

m X

(gj − g˜j + hj ) pj ,

j=1

and each term in the sum on the right hand side has total degree at most d − 1. If deg(f ) < d − 1 we can repeat the above argumentation, until we end up with the desired H–representation. This proves the implication ii) ⇒ i). Of course, condition ii) of theorem 4.2 is already satisfied if m X i=1

gi fi

∗ −→ 0

F

holds only for all elements (g1 , .., gm ) of a basis of Syz (M (H) (f1 ), . . . , M (H) (fm )). (By lemma 3.2 ii) we may assume, that every basis element belongs to a space Sd (M (H) (f1 ), . . . , M (H) (fm )).) Thus, theorem 4.2 suggests thus the following procedure for computing H– bases. Given a set of polynomials F = {f1 , . . . , fm } generating an ideal I, first the elements of a basis of Syz (M (H) (f1 ), . . . , M (H) (fm ))

(4.2)

have to be computed. Then for each basis element (g1 , . . . , gm ) the polynomial p as given in theorem 4.2 ii) has to be reduced modulo F as far as possible. If the reduction procedure terminates with a nonzero polynomial f , the set F is enlarged by f . For the new set F a basis for the module of syzygies is computed etc. This procedure differs from Buchberger’s algorithm only by the module of


13

syzygies. Instead of a basis of (4.2) one considers for Gröbner basis computation the basis

l.c.m.{lt(fi ), lt(fj )} l.c.m.{lt(fi ), lt(fj )} ei − ej | 1 ≤ i < j ≤ m lt(fi ) lt(fi )

(4.3)

of the module Syz (lt(f1 ), . . . , lt(fm )), where ei denotes the i-th unit vector and lt(fi ) is the leading term (maximal part) of fi , see for instance [1, p. 119]. As Buchberger’s algorithm terminates correctly with a Gr¨ obner basis, our proposed algorithm terminates with an H–basis. This procedure looks very similar to the one Macaulay presented in his example. However, its efficiency depends on the complexity of computing bases for modules of syzygies (4.2). There are a few instances where a construction of such such bases is known. First, if the M (H) (fi ) are just power products, then (4.3) is a basis. Secondly, if M (H) (f1 ), . . . , M (H) (fm ) are arbitrary (homogeneous) polynomials, but if a Gr¨ obner basis for the ideal hM (H) (f1 ), . . . , M (H) (fm )i is known, then a basis for (4.2) can be computed as described by Adams and Loustaunau in [1, p. 134ff], see also [7]. One can use the basis elements of one module of syzygies for the next module if the set F is enlarged by one element. However, this updating by Gr¨ obner techniques means, as the exposition in [1, p. 134ff] shows, that a Gr¨ obner basis for the given ideal is computed as a byproduct. But this Gr¨ obner basis is already an H–basis, if the ordering is degree compatible. Therefore it seems to be more efficient, to compute an H–basis rather by Buchberger’s algorithm using a degree compatible ordering than by the procedure sketched above and using Gr¨ obner techniques for the calculation of modules of syzygies. But there is an instance, where a basis for the module of syzygies Syz(f1 , . . . , fm ) can be described easily, namely if the polynomials f1 , . . . , fm constitute a regular sequence.

5.

Regular sequences

Definition 5.1. Let p1 , . . . , ps ∈ P. If i) hp1 , . . . , ps i = 6 P ii) pk is not a zerodivisor of P/hp1 , . . . , pk−1 i for k = 1, . . . , s, then {p1 , . . . , ps } is called a regular sequence.

14


One property of these sequences is that any nonempty subset of a regular sequence is a regular sequence as well. Such regular sequences are considered in modern textbooks on commutative algebra for instance in [10] and, also under the name A-sequence, in [11]. But even these sequences were already investigated by Macaulay. The ideal which is generated by a regular sequence is called nowadays complete intersection ideal, whereas Macaulay used in [14] the name module of the principal class. Lemma 5.2. Let {p1 , . . . , ps } be a regular sequence in P. Then the module of syzygies Syz (p1 , . . . , ps ) is generated by the syzygies pk ei − pi ek ,

1 ≤ i < k ≤ s,

where ei denotes the i-th unit vector. Proof. Obviously pk ei − pi ek ∈ Syz (p1 , . . . , ps ) for 1 ≤ i < k ≤ s. Condition ii) of the definition means, that for arbitrary g ∈ P g · pk ∈ hp1 , . . . , pk−1 i =⇒ g ∈ hp1 , . . . , pk−1 i. Consider a syzygy (g1 , . . . , gk , 0, . . . , 0) ∈ Syz (p1 , . . . , ps ). Then gk p k = −

k−1 X

gj pj ∈ hp1 , . . . , pk−1 i.

j=1

Therefore gk ∈ hp1 , . . . , pk−1 i. Hence gk =

Pk−1 i=1

hi pi and

k−1 X

hi (pk ei − pi ek ) i=1 ∗ = (g1∗ , . . . , gk−1 , 0, . . . , 0)

(g1 , . . . , gk , 0, . . . , 0) +

∈ Syz (p1 , . . . , ps )

This (and Syz (p1 ) = h0i) allows an inductive argument. A handy criterion for deciding the regularity of a polynomial sequence needs the notion of dimension of ideals. It can be defined by the degree of the affine Hilbert polynomial, see [9], but usually it is defined for prime ideals by the length of maximal sequences of prime ideals. Equivalently, it can be defined geometrically by the dimension of varieties as explained by Cox, Little, O’Shea [9]. The variety of an ideal is the set of all points (with coordinates in the algebraic closure


15

of K) being common zeros of all polynomials in the ideal. This variety splits into irreducible varieties which are varieties of prime ideals, i.e. ideals I satisfying p · q ∈ I, p 6∈ I

=⇒ q ∈ I.

The maximal geometric dimension of such irreducible variety is the dimension of the ideal, dim(I). Special instances are dim(I) = 0 if and only if the polynomials in I have only finitely many zeros in common, and dim(I) = −1 if and only if the polynomials of I have no zero in common. Hilbert’s Nullstellensatz shows that the latter is equivalent to I = P. For regular sequences of homogeneous polynomials {p1 , . . . , ps }, Macaulay proved the unmixedness theorem, which says that every irreducible subvariety (in the sense above) has the same dimension n − s as the ideal I = hp1 , . . . , ps i and conversely, if an H-ideal is generated by s (homogeneous) polynomials and has dimension n − s, then these s polynomials are a regular sequence. Therefore, under favorable circumstances we can determine if a polynomial set is a regular sequence and get, by lemma 5.2, a basis of its module of syzygies. But there is a situation, where we can get even more. Theorem 5.3. Let F := {p1 , . . . , pn } ⊂ P. If the leading homogeneous polynomials M (H) (p1 ), . . . , M (H) (pn ) have only the point (0, . . . , 0) as common zero, then F is an H–basis. Proof. Introducing a new variable x0 , each pi can be homogenized to Pi := Φdeg(pi ) pi ∈ K[x0 , . . . , xn ]. Then M (H) (pi ) = Pi |x0 =0 , i = 1, . . . , n. Hence, by assumption, the polynomials P1 , . . . , Pn , x0 have just (0, . . . , 0) as common zero. This means, they generate an ideal of dimension 0. But we have here n + 1 variables and n + 1 polynomials. Hence they constitute a regular sequence. The same holds for their subset {P1 , . . . , Pn }. They generate an ideal J of dimension 1 and F · x0 ∈ J implies F ∈ J. Hence P1 , . . . , Pn generate the H-ideal associated to I = hp1 , . . . , pn i and {p1 , . . . , pn } is an H–basis by theorem 4.1. If an ideal I is generated by a regular sequence, then its variety is called a “complete intersection”. In this case, we know the Hilbert function explicitely. If p1 , . . . , pn are as in theorem 5.3 with di := deg(pi ), i = 1, . . . , n. Then P1 := Φd1 (p1 ), . . . , Pn := Φdn (pn ) constitute a regular sequence as the proof above

16


showed. The ideal J = hP1 , . . . , Pn i is an H-ideal and its Hilbert function satisfies the Hilbert-Poincaré series identity ∞ X

HJ (d)X d =

d=0

(1 − X d1 ) · · · (1 − X dn ) , (1 − X)n+1

as shown for instance by Fröberg in [11, p. 137] in exercise 7. Since HJ (d) is the number of monomials of degree d in n + 1 variables minus the maximal number of linearly independent degree d elements in J, HJ (d) =

!

d+n − dim(J ∩ Hd ), n

and since there is the one-to-one correspondence between polynomials of degree at most d in I = hp1 , . . . , pn i and homogeneous polynomials of degree d in J, HJ (d) = a HI (d) holds by lemma 3.4 iii). A simple manipulation of power series shows then the following. Lemma 5.4. Let F := {p1 , . . . , pn } be as in theorem 5.3 with di := deg(pi ) and let I = hp1 , . . . , pn i. Then the following power series identity holds ∞ X

HI (d)X d =

d=0

(1 − X d1 ) · · · (1 − X dn ) . (1 − X)n

Especially HI (d) = 1 for d = d1 + d2 + . . . + dn − n, HI (d) = 0 for d > d1 + d2 + . . . + dn − n, a H (d) = d · d · · · d for d ≥ d + d + . . . + d − n. 1 2 n 1 2 n I Example 5.5. Let f1 := x4 + 2x2 y 2 + y 4 − 1 and f2 := xy(x2 − y 2 ) as in Example 2.4. Then these polynomials constitute already an H–basis by theorem 5.3, since their maximal parts (x2 + y 2 )2 and xy(x2 − y 2 ) have no zero in common except (0, 0). Therefore, the Hilbert-Poincaré series gives for I = hf1 , f2 i ∞ X

d=0

HI (d)X d =

(1 − X 4 )(1 − X 4 ) = (1 + X + X 2 + X 3 )2 (1 − X)2

Hence HI (d) = d + 1 for d = 0, 1, 2, 3 and HI (d) = 7 − d for d = 4, 5, 6. For P d > 6 = deg(f1 ) + deg(f2 ) − 2 we have HI (d) = 0. And a HI (d) = di=0 HI (d) = 16 = deg(f1 ) deg(f2 ) for d ≥ 6.


6.

17

H–bases and normal forms

Let us now study the reduction process introduced in section 2 more carefully with the notions of section 3. For a given set F := {f1 , . . . , fs }, we defined the linear spaces Vd (f1 , . . . , fs ) by (3.1), and, after having fixed an inner product in P, their orthogonal complements are defined as Wd (f1 , . . . , fs ) := P (H) Vd (f1 , . . . , fs ). Then the reduction f

−→ f˜ =

F

f−

m X

gi fi ,

deg(f˜) < d,

i=1

can be performed if and only if M (H) (f ) ∈ Vd (f1 , . . . , fs ), because exactly then the homogeneous part of degree d on the right hand side can cancel. We generalize this reduction in two ways. We weaken the condition that the complete maximal part M (H) (f ) has to vanish and we admit that reductions are performed also for homogeneous components of lower degree. Definition 6.1. Let F := {f1 , . . . , fs } ⊂ P and let ϕd = ϕd (f1 , . . . , fs ) denote (H) the orthogonal projection of Pd onto Wd (f1 , . . . , fs ), d ∈ N0 . We define recursively a reduction. We say f ∈ P reduces fully to f˜ modulo F, if either there are (H) polynomials gi ∈ Pdeg(f )−deg(fi ) , i = 1, . . . , s, such that f˜ = f −

s X

gi fi ,

M (H) (f˜) = ϕdeg(f ) (f ),

i=1

M (H) (f )

M (H) (f˜)

or if = and f − M (H) (f ), the reductum of f , reduces fully modulo F to f˜ − M (H) (f˜), the reductum of f˜. Each polynomial f =: f˜0 of degree d can be fully reduced modulo an arbitrary finite polynomial set F. At a first step, we reduce f˜0 fully modulo F to a polynomial wd + f˜1 with wd = ϕd (f˜0 ) ∈ Wd (f1 , . . . , fs ) and f˜1 ∈ Pd−1 . Then we reduce f˜1 to wd−1 + f˜2 with wd−1 ∈ Wd−1 (f1 , . . . , fs ) and f˜2 ∈ Pd−2 etc. This procedure terminates after d reduction steps. The result is as follows. Lemma 6.2. Let F := {f1 , . . . , fs } ⊂ P and f ∈ Pd . Then there are polynomials gi with deg(gi ) ≤ deg(f ) − deg(fi ), i = 1, . . . , s, and homogeneous wk ∈ Wk (f1 , . . . , fs ) for k = 0, . . . , d, such that f = wd + wd−1 + . . . + w0 +

s X i=1

gi fi .

(6.1)

18


If F is an H–basis, the w0 , . . . , wd are uniquely determined by f (and by F and the inner product) for every f ∈ P. Proof. The existence of a representation (6.1) is obvious by the construction above. If F is an H–basis and if an f has a second representation (6.1) with wk0 in place of wk for k = 0, . . . , d, then take the maximal index m such that 0 . The difference of the two representations gives wm 6= wm m X

(wi − wi0 ) ∈ hFi .

i=0

0 wm

Since wm − is the maximal part of (wi − wi0 ) the H–basis property ii) of 0 ∈ V (f , . . . , f ), but by construction w − w 0 ∈ thm. 2.3 shows wm − wm m 1 s m m 0 = 0, a contradiction. Wm (f1 , . . . , fs ), hence wm − wm P

Definition 6.3. Let F := {f1 , . . . , fm } ⊂ P. For f ∈ P of degree d let P w0 , . . . , wd be obtained as in (6.1). Then dk=0 wk is called a completely reduced form of f with respect to F (and the inner product (·, ·) in P). If F is an H–basis, then the completely reduced form of f is also called normal form of f , NF(f, F) for short, and for arbitrary f ∈ P the polynomial RRF(f, F) := f − NF(f − M (H) (f ), F) is called the reductum reduced form of f (with respect to the H–basis F). The normal form and the reductum reduced form are both well defined, since the completely reduced form is unique by lemma 6.2 if F is an H–basis. We now try to find a standard H–basis of a given ideal I. Let F = {f1 , . . . , fs } be an H–basis of I. Then F˜ := {RRF(f1 , F), . . . , RRF(fs , F)}

(6.2)

is still an H–basis of I, since fi − M (H) (fi ) and NF(fi − M (H) (fi ), F) differ only by an element of I, such that RRF(fi , F) ∈ I, and both fi and RRF(fi , F) have the same maximal part. By construction, every element f ∈ F˜ consists in a maximal P part belonging to Vdeg(f ) (I), whereas its reductum belongs to d−1 k=0 Wk (I). For simplicity of notation suppose that fi = RRF(fi , F) for i = 1, . . . , s. By theorem 2.3 ii), F is an H–basis of I if and only if Vd (I) = Vd (f1 , . . . , fs ) holds for every d ∈ N0 . This means that Vd (I) is generated by all t · M (H) (fi ), t a power product (or monomial) of degree d − deg(fi ), i = 1, . . . , s. Let f1 , . . . , fk be the H–basis elements of degree less than d and f1 , . . . , f` those of degree at


19

most d; clearly, k ≤ `. Then Vd (I) = Vd (f1 , . . . , f` ) holds if and only if Vd−1 (I) = Vd−1 (f1 , . . . , fk ) and the degree d polynomials M (H) (fk+1 ), . . . , M (H) (f` ) generate, together with a basis of Vd (f1 , . . . , fk ) the space Vd (I). Hence, fk+1 , . . . , f` may be chosen such that M (H) (fk+1 ), . . . , M (H) (f` ) are mutually orthogonal (with respect to the inner product) and also orthogonal to Vd (f1 , . . . , fk ). The orthogonalization of the maximal parts against Vd (f1 , . . . , fk ) may introduce homogeneous components which do belong to a Vj (I), j < d, and not to Wj (I). Therefore, an additional transition to reductum reduced forms is necessary. Theorem 6.4. Let F := {f1 , . . . , fs } be an H–basis of an ideal I and let (·, ·) be an inner product of P. Then an H–basis G = {g1 , . . . , gr } of I can be constructed, such that (M (H) (gi ), tM (H) (gj )) = 0

(H)

for all j 6= i, t = xi11 · · · xinn ∈ Pdeg(gi )−deg(gj )

and gi = RRF(gi , G) for all i = 1, . . . , r. If deg(gi ) 6= deg(gj ) for i = 6 j, then the H–basis elements gi , i = 1, . . . , r, are uniquely determined up to a constant multiple. Proof. The construction of the H–basis G is described above. Uniqueness follows by the fact that the maximal part of each gi is unique up to a constant by the orthogonality conditions. The condition gi = RRF(gi , G) determines the remaining part gi − M (H) (gi ) uniquely, since every completely reduced form is unique in case of H–bases. If two of the polynomials of the H–basis G have same degree, then the basis is not uniquely determined. Take for instance the H–basis considered before, F = {f1 , f2 } with f1 := x4 + 2x2 y 2 + y 4 − 1 and f2 := xy(x2 − y 2 ). Both polynomials have degree 4. They are reductum reduced, because there are no H–basis elements of lower degree. Take as inner product the scalarproduct of the coefficient vectors. Then, in this example M (H) (f1 ), M (H) (f2 ) = 0. But for arbitrary b, c ∈ K with b2 + c2 = 1 also the polynomials g1 := bf1 + cf2 , g2 = −cf1 + bf2 are an H–basis of the same ideal, reductum reduced and orthogonal, (H) (H) i.e. M (g1 ), M (g2 ) = 0. To explain this statement, we denote by deg(G) = max deg(g) g∈G

20


the degree of a finite set G of polynomials. We also associate to G = {g1 , . . . , gr } the vector G = (g1 , . . . , gr ) ,

deg (gj ) ≤ deg (gj+1 ) ,

j = 1, . . . , r − 1.

Finally, for j = 0, . . . , deg(G), we write dj = # {g ∈ G : deg(g) = dj } for the number of elements of degree j in G. With this notation at hand, we obtain the following uniqueness result for H–bases. Theorem 6.5. Let F = {f1 , . . . , fr } and G = {g1 , . . . , gs } be two completely reduced H–bases for an ideal I. Then r = s, deg(F) = deg(G) as well as dj (F) = dj (G), j = 0, . . . , deg(F), and there exists an orthogonal matrix A ∈ Kr×r consisting of diagonal dj × dj blocks such that F = AG. Proof. Let j0 (F) := min {j : dj (F) 6= 0} and define j0 (G) respectively. Since G ⊂ I = hFi and since F is an H–basis, we observe that j0 (F) ≤ j0 (G). By symmetry, we also have the converse inequality j0 (G) ≤ j0 (F), i.e., j0 (G) = j0 (F) =: j0 . Since G is an H–basis for I, any of the polynomials fj , j = 1, . . . , dj0 (F), must be a linear combination of the polynomials gj , j = 1, . . . , dj0 , and, again by symmetry, vice versa. Hence, span {fj : j = 1, . . . , dj0 (F)} = span {gj : j = 1, . . . , dj0 (G)} . Because F and G are completely reduced, the polynomials on both sides of the above identity are linearly independent and thus dj0 (F) = dj0 (G). Hence, there exists a nonsingular matrix Aj0 such that Fj0 = Aj0 Gj0 ; moreover, writing F ⊗ G = [(fj , gk ) : j, k = 1, . . . , d] for the Gramian of two vectors F, G of polynomials with respect to the inner product (·, ·), we find that Idj0 ×dj0 = Fj0 ⊗ Fj0 = Aj0 (Gj0 ⊗ Gj0 ) ATj0 = Aj0 Idj0 ⊗dj0 ATj0 = Aj0 ATj0 , which proves that Aj0 is an orthogonal matrix. Proceeding inductively with jk (F) = min {j > jk−1 : dj (F) 6= 0} and jk (G) respectively, k = 1, 2, . . ., we can use essentially the same argument. Indeed, we consider, for k = 1, 2, . . ., the spaces spanned by

fj : j = djk−1 + 1, . . . , djk (F)

and

gj : j = djk−1 + 1, . . . , djk (G)


21

and assume that at least on of them is nonempty, say the first one. Since we are dealing with completely reduced H–bases, we observe that d

span fj : j = djk−1 + 1, . . . , djk (F) = I ∩

jk M

Wd f1 , . . . , fdjk−1 ,

d=0

but by induction, the right hand side equals d

I∩

jk M

Wd g1 , . . . , gdjk−1 = span fj : j = djk−1 + 1, . . . , djk (G) ,

d=0

from where we can apply exactly the same argumentation as above. As the proof shows, the matrix A which relates the vectors F and G has the following block diagonal structure: 



A0 0 . . . 0    0 A1 0    A= . , ..  ..  . 

0 0

Ad

Aj ∈ Rdj ×dj , j = 0, . . . , d = deg(F).



In particular, we now obtain theorem 6.4 as the special case where dj ∈ {0, 1} and the A is a diagonal matrix which carries the respective normalization factors.

7.

Computing polynomial zeros by Eigenvectors

There is an interesting method to compute the common zeros of a set of polynomials by means of Eigenvalues/Eigenvectors. Stetter first presented this method in [2] using resultants. In subsequent papers such as [20], Stetter’s method was combined with Gröbner basis techniques. But also H–bases in place of Gröbner bases can be used as well. Let us briefly explain this method. For details and proofs we refer to M¨ oller and Stetter [20]. Consider a set of polynomials {f1 , . . . , fm } with only finitely many common zeros (in the algebraic closure of the ground field). Let I be the ideal generated by f1 , . . . , fm . Then its variety V (I) consists of these finitely many common zeros. We fix an f ∈ P \ I. Denoting by [g] the equivalence class [g] := {h ∈ P | g − h ∈ I } ,

22


the map Ψf : [g] 7→ [f · g] is an endomorphism of P/I. Its Eigenvalues are {f (z)| z ∈ V (I)}. For arbitrary f, h ∈ P the identities Ψf +h = Ψf + Ψh and Ψf ·h = Ψf ◦ Ψh = Ψh ◦ Ψf hold. If p1 , . . . , pN are polynomials such that [p1 ], . . . , [pN ] constitute a basis of P/I, then the matrix associated to Ψf with respect this basis is called the multiplication table and denoted by Mf . An Eigenvector for the Eigenvalue f (z) of Mf is given by χ(z) := (p1 (z), . . . , pN (z))T .

(7.1)

Remark that this Eigenvector is independent of the chosen f ! If the degree 1 polynomials x1 , . . . , xn are among p1 , . . . , pN , then we can read off all n coordinates of the common zero z ∈ V (I) from χ(z). The construction of a basis [p1 ], . . . , [pN ] is done in [20] by Gr¨ obner bases i1 i n techniques. The set of all power products x1 · · · xn which are not divisible by the leading terms of the polynomials of a Gr¨ obner basis G of I constitute the so called normal set N (G). The elements of this normal set are taken as p1 , . . . , pN . The (Gr¨ obner) normal form NF(·, G) maps onto the linear space spanned by N (G). Hence, the entries mij of Mf can easily been computed by reducing P pi f to NF(pi f, G) = N j=1 mij pj , i = 1, . . . , N . With an H–basis F of I at hand, one can proceed similarly. Every reduction modulo F remains within an equivalence class [g]. Hence the (H–basis) normal form NF(g, F) is a minimal total degree polynomial in [g] and we may take this polynomial as the standard representer of the equivalence class [g]. P Here, NF maps onto ∞ d=0 Wd (I). Since the variety V (I) is finite, the ideal I has dimension 0. Therefore the affine Hilbert polynomial of I is a degree 0 polynomial, i.e. a constant. Hence, there is a T = T (I) ∈ N such that HI (t) = P dim Wt (I) = 0 for all t ≥ T . This means that NF maps onto Td=0 Wd (I), a finite dimensional vector space. Its dimension equals the constant N := a HI (T ). Let P p1 , . . . , pN denote a basis of Td=0 Wd (I). Then {[p1 ], . . . , [pN ]} is a basis of P/I and the entries mij of a multiplication table Mf can be computed as before by P reducing pi f to NF(pi f, F) = N j=1 mij pj , i = 1, . . . , N . The Eigenvector χ(z) for z ∈ V (I) as given in 7.1 contains all n coordinates of z if x1 , . . . , xn are among the pi . This holds if and only if dim W1 (I) = n. Otherwise, I contains some linear polynomials ` ∈ P1 , such that by `(z) = 0 the


23

missing coordinate values of z can be reconstructed. Example 7.1. We continue the example 5.5 . f1 = x4 + 2x2 y 2 + y 4 − 1 and f2 = xy(x2 − y 2 ) constitute an H–basis of I = hf1 , f2 i. Therefore Wd (I) = Wd (f1 , f2 ) for all d ∈ N and, as already shown,

dim Wd (I) = HI (d) =

   d + 1   

7−d 0

for d = 0, 1, 2, 3, for d = 4, 5, 6, 7, for d ≥ 7 . (H)

We have hence T = T (I) = 7 and N = 16 and especially Wd (I) = Pd for d = 0, 1, 2, 3 and Wd (I) = h0i for d ≥ 7. If we take as inner product in P the scalar product of the coefficient vectors, then, as easy computation shows, the three dimensional space W4 (f1 , f2 ) orthogonal to V4 (f1 , f2 ) is generated by ϕ1 = x4 − y 4 , ϕ2 = x4 − x2 y 2 + y 4 , ϕ3 = x3 y + xy 3 . The two dimensional space W5 (f1 , f2 ) has as basis ϕ4 = −3x5 + x3 y 2 + xy 4 , ϕ5 = x4 y + x2 y 3 − 3y 5 . And the one dimensional space W6 (f1 , f2 ) is generated by ϕ6 = −3x6 + x4 y 2 + x2 y 4 − 3y 6 . Thus we may take as (p1 , . . . , pN ) the vector (1; x, y; x2 , xy, y 2 ; x3 , x2 y, xy 2 , y 3 ; ϕ1 , ϕ2 , ϕ3 ; ϕ4 , ϕ5 ; ϕ6 )T ,

(7.2)

where we used a semicolon in place of the usual separating comma in order to separate homogeneous polynomials of different degrees. If we consider the endomorphism Ψx then its matrix Mx corresponding to the given basis has entries mij , where NF(x · pi ) =

16 X

j=1

mij pj , i = 1, . . . , 16.

24


Computation gives 



1 0

  1 0    0 1          1 6  0 Mx =  1 3  0   1  − 11 0  2   11 0  6  0 11   − 27  20 0   0 − 14 

0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 2

1 3

0 0 0 − 13 0 0

33 20

0 −3 0 2 0

               0   1  2  .  0   1   2  4  − 11 0  3  − 11 0   2 0 11   11  20   0 

Here we separated by lines the entries corresponding to polynomials of different degrees. Empty blocks stand for blocks with only 0 entries. Since there are four common zeros z = (x, y) with x = 0, the Eigenspace of Mx corresponding to the Eigenvalue 0 contains four Eigenvectors of type (7.2). Every vector of that Eigenspace can be written as (b; 0, c; 0, 0, a; 0, 0, 0, d; −b, b, 0; 0, −3c; −3a)T

a, b, c, d arbitrary .

Comparing the entries with χ(0, y) = (1; 0, y; 0, 0, y 2 ; 0, 0, 0, y 3 ; −y 4 , y 4 , 0; 0, −3y 5 ; −3y 6 )T gives the necessary conditions b = 1, c = y, a = y 2 , d = y 3 ,

and y 4 = 1 .

These condition are also sufficient, since indeed every fourth unit root y ∈ {1, i, −1, −i} gives a common zero of type (0, y) for f1 and f2 . The remaining common zeros of f1 , f2 can be found by similar arguments in the other Eigenspaces of Mx . Here every Eigenspace has dimension greater than 1, such that every times a reconstruction as described is needed. The same holds


for the matrix My . But for every α ∈ C with at most matrix

16 2

25

= 120 exceptions, the

Mx+αy = Mx + αMy has only Eigenspaces of dimension 1 because for each of the 16 common zeros z = (x, y) of f1 and f2 the Eigenvalues x + αy are different if and only if no two of them are on the same line with normal vector (α, −1). For instance let α = 2. Then 

Mx+2y

1

              1 6 2  3 = 1 3 1  3  1  − 11  2   11  12  11      



2 1 0

2 1

0 2 1 2 0 0 0 1 2 0 0 0 1 2 1 2

1 3 0 − 23 0 − 13 −1 23

1 1 2

1 1 2

2 11 4 11 6 11

4 8 − 11 11 3 6 − 11 − 11 4 11

2 11

1 33 − 27 20 − 2 20 33 1 27 − − 10 4 10

−3 4 2 −6

                   .              11  20   11  10 

All its Eigenspaces are one dimensional. Computation gives Eigenvalue ±2i (1; ±i (1; ±1 (1; ±2 (1; ±i √ (1; 2 ±1 √ 2 ±3i √ 2 ±3 √ 2

(1; (1; (1;

0, ±i ; ±i, 0 ; ±1, 0 ; 0, ±1 ; ∓i ±i √ ,√ ; 2 2 ∓1 √ , 2 ±i √ , 2 ±1 √ , 2

±1 √ 2 ±i √ 2 ±1 √ 2

; ; ;

Eigenvector 0, 0, −1; 0, 0, 0, ∓i; −1, 1, 0; 0, 3; 3) −1, 0, 0; ∓i, 0, 0, 0; 1, 1, 0; ∓3i, 0; 3) 1, 0, 0; ±1, 0, 0, 0; 1, 1, 0; ∓3, 0; −3) 0, 0, 1; 0, 0, 0, ±1; −1, 1, 0; 0, ∓3; −3) −1 1 −1 √ ±1 √ ±1 √ ±1 √ ∓1 1 √ , ∓18 , √ , ∓18 ; 0, 14 , −1 2 , 2, 2 ; 2 ; 32 , 32 ; 2 ) 8 8 1 −1 1 2, 2 , 2; −1 −1 −1 2 , 2 , 2 ; 1 1 1 2, 2, 2;

∓1 √ , 8 ∓i √ , 8 ±1 √ , 8

±1 √ , 8 ∓i √ , 8 ±1 √ , 8

∓1 √ , 8 ∓i √ , 8 ±1 √ , 8

∓1 √ ; 0, 14 , −1 2 ; 8 ∓i 1 1 √ ; 0, 4 , 2 ; 8 ±1 √ ; 0, 1 , 1 ; 4 2 8

±1 √ , 32 ∓i √ , 32 ∓1 √ , 32

∓1 √ ; 32 ∓i √ ; 32 ∓1 √ ; 32

−1 2 ) 1 2) −1 2 )

26


The Eigenvectors with 1 as first entry are the Eigenvectors χ(z), z ∈ V (I), as given in (7.2). Hence their second and third entry are the x- and y-coordinates of the common zeros z.

8.

Multivariate polynomial interpolation

In this section we point out how H–bases can be used minimal n to determine o 0 N degree interpolation spaces. For this purpose, let X = x , . . . , x ⊂ Kn be a set of N + 1 distinct points, N ≥ 0, and consider the problem of finding, for given data y0 , . . . , yN ∈ K, an interpolation polynomial p ∈ P such that p (xj ) = yj ,

j = 0, . . . , N.

(8.1)

Clearly, this problem has many solutions and all solutions of the homogeneous problem y0 = · · · = yN = 0 form an ideal I ⊂ P. Let, for f ∈ P, the equivalence class modulo I be denoted by [f ] = {p ∈ K [x1 , . . . , xn ] | p − f ∈ I} . If for given y0 , . . . , yN ∈ K both p, q ∈ P are solutions of (8.1), then p − q ∈ I or [p] = [q]. We also remark that the quotient space P/I is finite dimensional and that dim P/I = N + 1. We want to construct solutions of (8.1) which are of smallest possible degree. Hence, if p is any solution of the interpolation problem (8.1), then we choose the interpolant L (y0 , . . . , yN ) ∈ [p] so that deg L (y0 , . . . , yN ) ≤ deg q,

q ∈ [p].

(8.2)

Note that there always exists at least one solution of (8.1) of minimal degree and that the selection in (8.2) can be performed for any grading structure. However, the choice of the interpolation polynomial is very vague so far, but we will show that there even exists a linear interpolation operator which produces solutions of (8.1). Of course, the above operator can easily be extended to P by setting

Lp = L p x0 , . . . , p xN

,

p ∈ P,

(8.3)

which immediately leads to the following observation. Lemma 8.1. The operator L : P → P defined in (8.3) is degree reducing, i.e., deg Lp ≤ deg p for all p ∈ P


27

Our goal is to use H–bases to construct linear interpolation operators which are degree reducing. For that purpose, we make the following definition. Definition 8.2. A subspace P ⊂ P is called a minimal degree interpolation space with respect to X if 1. for any y0 , . . . , yN ∈ K there exists a unique polynomial LP (y0 , . . . , yN ) ∈ P such that LP (y0 , . . . , yN ) (xj ) = yj ,

j = 0, . . . , N.

2. for any f ∈ P the respective interpolation polynomial LP f satisfies deg (LP f ) ≤ deg(f ). The second property, degree reduction, can obviously be extended to any “reasonable” notion of degree, i.e., to any grading as pointed out in section 2. Moreover, this property guarantees that the interpolation space P is of minimal degree in the sense that no polynomial subspace which consists only of polynomials of degree strictly less than the maximal degree appearing in P does not allow unique interpolation. It is remarkable that minimal degree interpolation spaces have already been proposed in 1976 in [16] which was even done in the algebraic context of H–bases, disguised as “kanonische Basen” at this time. Interpolation with respect to a minimal degree interpolation space (which has to have dimension N + 1 and therefore is a representer for P/I) is a linear operation which associates to any polynomial a minimal degree element of the same equivalence class modulo P/I. We begin by following [18] and take a detour via Gröbner bases with respect to a degree compatible term order. As remarked before, such a Gröbner basis is also an H–basis. We recall the following result. Proposition 8.3 (cf. [9]). If G ⊂ P is a Gröbner basis, then there exists, for ∗ any f ∈ P, a unique irreducible polynomial g ∈ P such that f −G→ g. So, let G be such a Gröbner basis for the ideal I and let RG : P → P be the reduction operator modulo G which associates to each f ∈ P the polynomial RG f := g, where g is the irreducible polynomial from proposition 8.3. The crucial observation is as follows.

28


Lemma 8.4. Let G be a Gröbner basis for the ideal I. Then RG is a linear operator P → P/I. Proof. Homogeneity, i.e., RG (af ) = aRG f , a ∈ K, is obvious. To show additivity, we let f1 , f2 ∈ P. Since irreducibility of a polynomial f ∈ P means that none of the leading terms {lt(g) | g ∈ G} divides any term of f , we can conclude that irreducibility is an additive property and, in particular, that RG f1 + RG f2 ∗ is irreducible. It remains to show that f1 + f2 −G→ RG f1 + RG f2 , which is clearly true if lt (f1 ) 6= lt (f2 ). But otherwise there are only two possibilities: i) f1 − lt (f1 ) ∈ I and f2 − lt (f2 ) ∈ I, ii) lt (f1 ) = lt (RG f1 ) and lt (f2 ) = lt (RG f2 ). In case i) we simply proceed with f1 − lt (f1 ) instead of f1 and f2 − lt (f2 ) instead of f2 while in case ii) we observe that a cancellation among the leading terms of f1 and f2 also implies a cancellation of the leading terms of RG f1 and RG f2 . In both cases however, comparison of degrees yields that deg (RG f1 + RG f2 ) < deg (f1 + f2 ) , ∗ −→

hence f1 + f2 G RG f1 + RG f2 . That the range of RP is P/I, is easy to see: whenever g ∈ [f ], then f − g reduces to zero, hence RP (f − g) = 0, i.e., RP g = RP f . Based on this lemma, we can now define a genuine minimal degree interpolation space. Theorem 8.5. Let G be a Gr¨ obner basis for the ideal IX . Then the set RG (P) is a minimal degree interpolation space with respect to X and the associated interpolation operator LP takes the form LP = RG . The interpolation space defined in Theorem 8.5 has been around for a long time in the context of constructing Gröbner bases of ideals from dual functionals or, in other words, interpolation conditions, cf. [8,18]. As pointed out for example in [21], the advantage of these interpolation spaces is that they only use #X monomials, hence they are “minimally supported”. Using our tools from sections 3 and 4, we can easily extend the above idea to interpolation with polynomials reduced modulo an H–basis F = {f1 , . . . , fs }, since lemma 6.2 provides us with a generalization of proposition 8.3. Here, the


29

reduction operator RF is the normal form operator from definition 6.3, NF( · , F), L which maps P onto d Wd (f1 , . . . , fs ) = RF (P). Therefore, theorem 8.5 has the following “H–analog”. Theorem 8.6. Let F be an H–basis for the ideal IX . Then the set RF (P) is a homogeneous minimal degree interpolation space with respect to X and the associated interpolation operator LF takes the form LF = RF . It is interesting that, for a special choice of the scalar product, this interpolation space has been investigated before, however, in a different context. This is the concept of “least interpolation”, introduced by de Boor and Ron in [5]. The scalar product they chose (for K = R, to be precise, otherwise some obvious complex conjugation has to be added) is (f, g) = (f (D)g) (0),

f (D) = f

∂ ∂ ,..., . ∂x1 ∂xn

(8.4)

It is easy to see that this bilinear form is indeed strictly positive definite and symmetric. The scalar product (8.4) can be extended to the case that one of the arguments is a formal power series, say g ∈ K (x1 , . . . , xn ). With respect to this scalar product, any functional λ ∈ P 0 can be represented by a formal power series fλ ∈ K (x1 , . . . , xn ) such that λ(p) = (fλ , p) ,

p ∈ P.

In particular, if λ = δx is a point evaluation functional, then we have that fλ = fx is the power series expansion of t 7→ ex·t . As any polynomial f has a well–defined maximal nonzero homogeneous term M (H) (f ), any formal power series f (and, in particular, any polynomial) has a well–defined minimal (or “least”) nonzero homogeneous term which we denote by m(H) (f ). Then, in the case of a Lagrange interpolation problem, the least interpolation space Pl is defined as n

o

Pl = m(H) (f ) : f ∈ span {fx0 , . . . , fxN } . And indeed, this approach is nothing but the homogeneous normal form interpolation defined above. Theorem 8.7 ([23]).Let F be an H–basis for the ideal IX . For the reduction process based on the scalar product (8.4) one has Pl = RF (P).

30


Proof. The proof is based on the characterization Pl =

\

{f ∈ P : p(D)f = 0} ,

(8.5)

p∈IX

which can be found in [6]. Hence, (p, f ) = 0 whenever p ∈ Pl and f ∈ IX and therefore Pl ⊆ RF (P). On the other hand, both the linear spaces are interpolation spaces and, consequently, have dimension N = #X . Hence, Pl = RF (P) Thus, using H–bases we have identified the “least interpolation” scheme as a remainder of division. The “least” property, on the other hand, is due to the particular inner product which is used for the reduction process. Moreover, since for Gr¨ obner basis computations the inner product is irrelevant (the homogeneous spaces are one–dimensional), one can expect a similar property to hold for these normal form interpolation spaces as well. And indeed, it has been shown it [22] that normal forms with respect to Gröbner bases correspond to a “term order least interpolation” scheme. In this respect, H–bases provided the useful tool to illuminate the parallels between these two seemingly different concepts of interpolation. References [1] W. W. Adams and P. Loustauneau, An Introduction to Gr¨ obner Bases, Graduate Studies in Math. 3, AMS, 1994. [2] W. Auzinger and H.J. Stetter, An elimination algorithm for the computation of all zeros of a system of multivariate polynomial equations, Intern. Series in Numer. Math., 86, (1988) 11 – 30. [3] T. Becker and V. Weispfenning, Gr¨ obner Bases: A Computational Approach to Commutative Algebra, Springer Verlag, Berlin and New York, 1993. [4] G. Birkhoff, The algebra of multivariate interpolation, in Constructive Approaches to Mathematical Models, C. V. Coffman and G. J. Fix eds., Academic Press, New York, 1979, pp. 345 – 363. [5] C. de Boor and A. Ron. On multivariate polynomial interpolation. Constr. Approx., 6, (1990) 287–302. [6] C. de Boor and A. Ron. The least solution for the polynomial interpolation problem. Math. Z., 210, (1992) 347–378. [7] B. Buchberger, Gr¨ obner bases: An algorithmic method in polynomial ideal theory, in Multidimensional Systems Theory, N. K. Bose, ed., D. Reidel Publishing Company, 1985, pp. 184–232. [8] B. Buchberger and H. M. M¨ oller, The construction of multivariate polynomials with preassigned zeros, Springer Lecture Notes in Comp. Sci., 144, (1982) 24 – 31.


31

[9] D. Cox, J. Little, and D. O’Shea, Ideals, Varieties, and Algorithms, Springer Verlag, New York, 1992. [10] D. Eisenbud, Commutative Algebra With a View Toward Algebraic Geometry, Springer Verlag, New York, 1995. [11] R. Fr¨ oberg, An Introduction to Gr¨ obner Bases, John Wiley & sons, Chichester, 1997. [12] W. Gr¨ obner, Moderne Algebraische Geometrie, Springer Verlag, Wien-Innsbruck, 1949. [13] W. Gr¨ obner, Algebraische Geometrie II, Bibliographisches Institut, Mannheim-WienZ¨ urich, 1970. [14] F. S. Macaulay. The Algebraic Theory of Modular Systems. Number 19 in Cambridge Tracts in Math. and Math. Physics. Cambridge Univ. Press, 1916. [15] F. S. Macaulay, Some properties of enumeration in the theory of modular systems, Proc. London Math. Soc., 26, (1927) 531 – 555. [16] H. M. M¨ oller. Mehrdimensionale Hermite–Interpolation und numerische Integration. Math. Z., 148 (1976), 107–118. [17] H. M. M¨ oller. On the construction of Gr¨ obner bases using syzygies. J. Symbolic Comput., 6 (1988), 345–359. [18] H. M. M¨ oller, Gr¨ obner bases and Numerical Analysis, In B. Buchberger, and F. Winkler, eds., Groebner Bases and Applications (Proc. of the Conf. 33 Years of Groebner Bases) , London Math. Soc. Lecture Notes vol. 251, Cambridge University Press, 159–178. [19] H. M. M¨ oller and F. Mora, New constructive methods in classical ideal theory, J. of Algebra, 100, (1986) 138 – 178. [20] H. M. M¨ oller and H. J. Stetter, Multivariate polynomial equations with multiple zeros solved by matrix eigenproblems. Numer. Math. 70, (1995), 311 – 329. [21] T. Sauer, Polynomial interpolation of minimal degree, Numer. Math., 78, (1997) 59–85. [22] T. Sauer, Polynomial interpolation of minimal degree and Gr¨ obner bases. In B. Buchberger, and F. Winkler, eds., Groebner Bases and Applications (Proc. of the Conf. 33 Years of Groebner Bases) , London Math. Soc. Lecture Notes vol. 251, Cambridge University Press, 483–494. [23] T. Sauer, Gr¨ obner bases, H–bases and interpolation, Proc. Amer. Math. Soc., 1999, to appear.

Hâbases for polynomial interpolation and system solving

Hâbases for polynomial interpolation and system solving

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