The lex game and some applications

Felszeghy, Bálint; Ráth, Balázs; Rónyai, Lajos

doi:10.1016/j.jsc.2005.11.003

Cited by 44 publications

(68 citation statements)

References 6 publications

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“…, ≺ n . We claim that (the general form of) Theorem 9 together with the universality property of standard monomials (see [6]) prove the same result for arbitrary fields. For this we remark, that the proof of Theorem 9 uses only the elimination property of lex orders and the fact that the number of standard monomials of the ideal considered is the same for every term order.…”

Section: Theorem 9 ([10])mentioning

confidence: 55%

“…Standard monomials have some very nice properties; among other things, they form a linear basis of the F-vector space F[x]/I and in the case of vanishing ideals of finite set systems they are all square-free monomials. In general for vanishing ideals of finite vectors sets, not merely 0 − 1 vectors, their number equals the size of the defining point set and for lex orders they can be computed in linear, O(n|F|k) time, where k is the number of different coordinates appearing (see [6]). …”

Section: Algebraic Description Of S-extremal Familiesmentioning

confidence: 99%

“…. , a k } n ⊆ F n (see the universality property of standard monomials in [6]). For 1 ≤ i ≤ n, the i-section of V ⊆ {0, 1, .…”

Section: Extremal Vector Systemsmentioning

confidence: 99%

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Standard Monomials and Extremal Vector Systems

Mészáros

Rónyai

2017

Electronic Notes in Discrete Mathematics

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We say that a set system F ⊆ 2[n] shatters a given set S ⊆ [n] if 2 S = {F ∩ S : F ∈ F }. The Sauer-Shelah lemma states that in general, a set system F shatters at least |F| sets. A set system is called shattering-extremal if it shatters exactly |F| sets. In [9] and [13] an algebraic characterization of shattering-extremal set systems was given, which offered the possibility to generalize the notion of extremality to general finite vector systems. Here we generalize the results obtained for set systems to this more general setting, and as an application, strengthen a result of Li, Zhang and Dong from [8].

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Section: Theorem 9 ([10])mentioning

confidence: 55%

Section: Algebraic Description Of S-extremal Familiesmentioning

confidence: 99%

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Standard Monomials and Extremal Vector Systems

Mészáros

Rónyai

2017

Electronic Notes in Discrete Mathematics

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“…Following [8] and [14] we recall some facts about the Lex game, a method to determine the lexicographic standard monomials of the vanishing ideal of a finite set of points from F n , where F is an arbitrary field. Let V ⊆ F n be a finite set, and w = (w 1 , .…”

Section: Lex Standard Monomials For Linear Sperner Systemsmentioning

confidence: 99%

A note on linear Sperner families

Hegedüs

Rónyai

2018

Algebra Univers.

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In an earlier work we described Gröbner bases of the ideal of polynomials over a field, which vanish on the set of characterisic vectors v ∈ {0, 1} n of the complete d unifom set family over the ground set [n]. In particular, it turns out that the standard monomials of the above ideal are ballot monomials. We give here a partial extension of this fact. A set family is a linear Sperner system if the characteristic vectors satisfy a linear equation a 1 v 1 + · · · + a n v n = k, where the a i and k are positive integers. We prove that the lexicographic standard monomials for linear Sperner systems are also ballot monomials, provided that 0 < a 1 ≤ a 2 ≤ · · · ≤ a n . As an application, we confirm a conjecture of Frankl in the special case of linear Sperner systems. 2010

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“…Ideal theoretic methods, in particular Gröbner basis methods, are being widely applied to several problems in Science and Engineering, see [4]. More specifically, we can find their applications to Combinatorics in [5], [6], [7], [9]. In this paper, we introduce these methods to study difference sets.…”

Section: Introductionmentioning

confidence: 99%

Polynomial Criterion for Abelian Difference Sets

Keskar

Kumari

2020

Indian J Pure Appl Math

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Difference sets are subsets of a group satisfying certain combinatorial property with respect to the group operation. They can be characterized using an equality in the group ring of the corresponding group. In this paper, we exploit the special structure of the group ring of an abelian group to establish a one-to one correspondence of the class of difference sets with specific parameters in that group with the set of all complex solutions of a specified system of polynomial equations. The correspondence also develops some tests for a Boolean function to be a bent function.

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The lex game and some applications

Cited by 44 publications

References 6 publications

Standard Monomials and Extremal Vector Systems

Standard Monomials and Extremal Vector Systems

A note on linear Sperner families

Polynomial Criterion for Abelian Difference Sets

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