Polynomial partitioning for several sets of varieties

POLYNOMIAL PARTITIONING FOR SEVERAL SETS OF VARIETIES

arXiv:1601.01629v1 [math.AT] 7 Jan 2016

´ ALEKSANDRA S. DIMITRIJEVIC ´ BLAGOJEVIC, ´ AND GUNTER ¨ PAVLE V. M. BLAGOJEVIC, M. ZIEGLER Abstract. We give a new, systematic proof for a recent result of Larry Guth and thus also extend the result to a setting with several families of varieties: For any integer D ≥ 1 and any collection of sets Γ1 , . . . , Γj of low-degree k-dimensional varieties in Rn there exists a non-zero polynomial p ∈ R[X1 , . . . , Xn ] of degree at most D so that each connected component of Rn \Z(p) intersects O(jD k−n |Γi |) varieties of Γi , simultaneously for every 1 ≤ i ≤ j. For j = 1 we recover the original result by Guth. Our proof, via an index calculation in equivariant cohomology, shows how the degrees of the polynomials used for partitioning are dictated by the topology, namely by the Euler class being given in terms of a top Dickson polynomial.

1. Introduction The celebrated work by Guth and Katz [5] on the Erd˝ os distinct distances problem in the plane brought to light the following beautiful partitioning result: Theorem 1.1 (Guth and Katz 2015 [5, Thm. 4.1]). Let X be a finite set of points in Rn , and let D ≥ 1 be an integer. Then there exists a non-zero polynomial p ∈ R[X1 , . . . , Xn ] of degree at most D such that each connected component of the complement Rn \Z(p) contains at most Cn D−n |X| points of X, where Cn is a constant that may depend on n. Here Z(p) denotes the set of zeroes in Rn of the polynomial p, that is Z(p) = {(x1 , . . . , xn ) ∈ Rn : p(x1 , . . . , xn ) = 0}. In his recent paper [4], Guth used equivariant topology to prove the following extended polynomial partitioning result. Theorem 1.2 (Guth, 2015 [4, Thm. 0.3]). Let Γ be a finite set of k-dimensional varieties in Rn , each of them defined by at most m polynomial equations of degree at most d. Then for any D ≥ 1 there exists a non-zero polynomial p ∈ R[X1 , . . . , Xn ] of degree at most D such that each connected component of the complement Rn \Z(p) intersects at most C(d, m, n)Dk−n |Γ| varieties in Γ, where C(d, m, n) is a constant that may depend on the parameters d, m, and n. In this paper, based on the set-up from the proof of the previous theorem and the use of the Fadell– Husseini index theory [3], we make the next extension step by proving the following “colored” generalization of Theorem 1.2. Theorem 1.3. Let j ≥ 1 be an integer. For 1 ≤ i ≤ j, let Γi be a finite set of ki -dimensional varieties in Rn , each of them defined by at most mi polynomial equations of degree at most di . Then for any D ≥ 1 there exists a non-zero polynomial p ∈ R[X1 , . . . , Xn ] of degree at most D such that each connected component of the complement Rn \Z(p) for every 1 ≤ i ≤ j intersects at most C(di , mi , n)jDki −n |Γi | varieties in Γi , where C(di , mi , n) is a constant that may depend on parameters di , mi , and n. In a concrete example, this says the following: There are constants C1 = C(1, 2, 3) and C2 = C(1, 3, 3) such that if we have (large) collections Γ1 of red lines and Γ2 of blue points in R3 , then for every D ≥ 1 there is a nonzero polynomial p(x, y, z) ∈ R[x, y, z] of degree at most D such that each connected |Γ2 | |Γ1 | component of R3 \ Z(p) meets at most 2C1 2 red lines, and at most 2C2 3 blue points. (This is the D D special case when we have j = 2 families of varieties in R3 , so n = 3, where the first family consists of lines, so k1 = 1 and e.g. m1 = 2, d1 = 1, and the second one of points, so k2 = 0, m2 = 3, d2 = 1.) The research by Pavle V. M. Blagojevi´ c leading to these results has received funding from DFG via Berlin Mathematical School. Also supported by the grant ON 174008 of the Serbian Ministry of Education and Science. The research by Aleksandra Dimitrijevi´ c Blagojevi´ c leading to these results has received funding from the grant ON 174008 of the Serbian Ministry of Education and Science. The research by G¨ unter M. Ziegler received funding from DFG via the Research Training Group Methods for Discrete Structures and the Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics.” 1

´ DIMITRIJEVIC ´ BLAGOJEVIC, ´ AND ZIEGLER BLAGOJEVIC,

2

2. Proof of Theorem 1.3 The proof of Theorem 1.3 will have several separate components that at the end of the proof merge into the final argument. We also rely on two particular results by Solymosi and Tao [7, Thm. A.2] and by Guth [4, Lemma 3.1]. 2.1. Let Pnδ be the vector space of polynomials in n variables of degree at most δ with real coefficients, Pnδ = {p ∈ R[X1 , . . . , Xn ] : deg p ≤ δ}.
δn The dimension of this vector space is dim Pnδ = δ+n > n! . For every integer ℓ ≥ 1 choose the smallest n integer δℓ with the property that j2ℓ−1 ≤

δℓn n!

< j2n 2l−1 ,

or equivalently 1

1

(n!) n j n 2 In particular,

dim Pnδℓ

> j2

ℓ−1

ℓ−1 n

1

1

≤ δℓ < 2(n!) n j n 2

ℓ−1 n

.

(1)

. Next, let s be the smallest integer such that s X

δℓ ≤ D