More results on the number of zeros of multiplicity at least <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml48" display="inline" overflow="scroll" altimg="si27.gif"><mml:mi>r</mml:mi></mml:math>

Geil, Olav; Thomsen, Casper

doi:10.1016/j.disc.2017.01.004

Cited by 3 publications

(5 citation statements)

References 10 publications

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“…The codes obtained in this way are called affine cartesian codes. Affine cartesian codes were defined in [10] and further studied in, for example, [1,12,8,2]. In [10,Theorem 3.8] the authors determined the minimum distance of these codes.…”

Section: Affine Cartesian Codes and Their Higher Weightsmentioning

confidence: 99%

Generalized Hamming weights of affine Cartesian codes

Beelen

Datta

2018

Finite Fields and Their Applications

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In this article, we give the answer to the following question: Given a field F, finite subsets A 1 , . . . , Am of F, and r linearly independent polynomials f 1 , . . . , fr ∈ F[x 1 , . . . , xm] of total degree at most d. What is the maximal number of common zeros f 1 , . . . , fr can have in A 1 ×· · ·×Am? For F = Fq, the finite field with q elements, answering this question is equivalent to determining the generalized Hamming weights of the so-called affine Cartesian codes. Seen in this light, our work is a generalization of the work of Heijnen-Pellikaan for Reed-Muller codes to the significantly larger class of affine Cartesian codes.

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Section: Affine Cartesian Codes and Their Higher Weightsmentioning

confidence: 99%

Generalized Hamming weights of affine Cartesian codes

Beelen

Datta

2018

Finite Fields and Their Applications

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“…Now, since V J (I) = V J (I(V J (I); J )) by Proposition 1, and equality holds in (12) for I, we have that…”

Section: Sharpness and Equality Conditionsmentioning

confidence: 95%

“…for all ideals I ⊆ F[x]. If equality holds in (12) for a given ideal I ⊆ F[x], then equality holds in (4) for such ideal.…”

Section: Sharpness and Equality Conditionsmentioning

confidence: 99%

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

Geil¹,

Martínez-Peñas²

2018

Combinator. Probab. Comp.

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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...

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“…The interested reader can find the details in the arXiv version of this paper [5]. [21,Prop. 17] shows that equality holds in Theorem 7.11 when R is a field and f is polylinear (see Section 3.4).…”

Section: Multiplicitiesmentioning

confidence: 99%

On Zeros of a Polynomial in a Finite Grid

Bishnoi¹,

Clark²,

Potukuchi³

et al. 2018

Combinator. Probab. Comp.

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A 1993 result of Alon and Füredi gives a sharp upper bound on the number of zeros of a multivariate polynomial over an integral domain in a finite grid, in terms of the degree of the polynomial. This result was recently generalized to polynomials over an arbitrary commutative ring, assuming a certain "Condition (D)" on the grid which holds vacuously when the ring is a domain. In the first half of this paper we give a further Generalized Alon-Füredi Theorem which provides a sharp upper bound when the degrees of the polynomial in each variable are also taken into account. This yields in particular a new proof of Alon-Füredi. We then discuss the relationship between Alon-Füredi and results of DeMillo-Lipton, Schwartz and Zippel. A direct coding theoretic interpretation of Alon-Füredi Theorem and its generalization in terms of Reed-Muller type affine variety codes is shown which gives us the minimum Hamming distance of these codes. Then we apply the Alon-Füredi Theorem to quickly recover -and sometimes strengthen -old and new results in finite geometry, including the Jamison/Brouwer-Schrijver bound on affine blocking sets. We end with a discussion of multiplicity enhancements.

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More results on the number of zeros of multiplicity at least r

Cited by 3 publications

References 10 publications

Generalized Hamming weights of affine Cartesian codes

Generalized Hamming weights of affine Cartesian codes

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

On Zeros of a Polynomial in a Finite Grid

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