Bounding the Number of Common Zeros of Multivariate Polynomials and Their Consecutive Derivatives

Abstract

We upper bound the number of common zeros over a finite grid of multivariate polynomials and an arbitrary finite collection of their consecutive Hasse derivatives (in a coordinate-wise sense). To that end, we make use of the tool from Gröbner basis theory known as footprint. Then we establish and prove extensions in this context of a family of well-known results in algebra and combinatorics. These include Alon's combinatorial Nullstellensatz [1], existence and uniqueness of Hermite interpolating polynomials over a grid, estimations on the parameters of evaluation codes with consecutive derivatives [19], and bounds on the number of zeros of a polynomial by DeMillo and Lipton [7], Schwartz [24], Zippel [25,26], and Alon and Füredi [2]. As an alternative, we also extend the Schwartz-Zippel bound to weighted multiplicities and discuss its connection with our extension of the footprint bound.If V ≥r (F (x)) denotes the set of zeros of F (x) of multiplicity at least r, then a weaker, but still sharp, bound is the following:( 2) In the multivariate case, the standard approach is to consider the first r consecutive Hasse derivatives as those whose multiindices have order less than r, where the order of a multiindex (i 1 , i 2 , . . . , i m ) is defined as m j=1 i j . We will use the terms standard multiplicities to refer to this type of multiplicities. In this work, we consider arbitrary finite families J of multiindices that are consecutive in a coordinate-wise sense: if (i 1 , i 2 , . . . , i m ) belongs to J and k j ≤ i j , for j = 1, 2, . . . , m, then (k 1 , k 2 , . . . , k m ) also belongs to J . Obviously, the (finite) family J of multiindices of order less than a given positive integer r satisfies this property, hence is a particular case.Our main contribution is an upper bound on the number of common zeros over a grid of a family of polynomials and their (Hasse) derivatives corresponding to a finite set J of consecutive multiindices. This upper bound makes use of the technique from Gröbner basis theory known as footprint [10,15], and can be seen as an extension of the classical footprint bound [6, Section 5.3] in the sense of (2). A first extension for standard multiplicities has been given as Lemma 2.4 in the expanded version of [23].We will then show that this bound is sharp for ideals of polynomials, characterize those which satisfy equality, and give as applications extensions of known results in algebra and combinatorics: Alon's combinatorial Nullstellensatz [1,3,5,20,22], existence and uniqueness of Hermite interpolating polynomials [9,18,21], estimations on the parameters of evaluation codes with consecutive derivatives [11,18,19], and the bounds by DeMillo and Lipton [7], Zippel [25,26], and Alon and Füredi [2], and a particular case of the bound given by Schwartz in [24, Lemma 1].The bound in [24, Lemma 1] can also be derived by those given by DeMillo and Lipton [7], and Zippel [25, Theorem 1], [26, Proposition 3] (see Proposition 3 below), and is referred to as the Schwartz-Zippel bound in many works i...

Subspace coverings with multiplicities

“…That is, we shall seek the minimum number of hyperplanes in F n needed to cover the nonzero points at least k times, while covering the origin fewer times. Previous work in this direction has imposed the stricter condition of avoiding the origin altogether; Bruen [8] considered this problem over finite fields, while Ball and Serra [4] and Kós and Rónyai [19] worked with finite grids over arbitrary fields, with some further generalisations recently provided by Geil and Matrínez-Penas [12]. In all of these papers, the polynomial method described above was strengthened to obtain lower bounds for this problem with higher multiplicities.…”

Subspace coverings with multiplicities

“…That is, we shall seek the minimum number of hyperplanes needed in F n to cover the nonzero points at least k times, while the origin is covered fewer times. Previous work in this direction has imposed the stricter condition of avoiding the origin altogether; Bruen [8] considered this problem over finite fields, while Ball and Serra [4] and Kós and Rónyai [16] worked with finite grids over arbitrary fields, with some further generalisations recently provided by Geil and Matrínez-Peñas [11]. In all of these papers, the polynomial method described above was strengthened to obtain lower bounds for this problem with higher multiplicities.…”

Polynomials over structured grids

“…By the degree of a multivariate polynomial we always mean its total degree. Algebraic aspects of the broader theme, polynomials over finite grids, have been investigated in many recent papers [3,5,7,12,16,22]. We may also refer to [15,19] for alternate approaches to the Combinatorial Nullstellensatz.…”