有限域上的多仿射多项式

IF 0.3 Q4 MATHEMATICS, APPLIED

Discrete Mathematics and Applications Pub Date : 2021-12-01 DOI:10.1515/dma-2021-0038

S. Selezneva

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引用次数: 1

摘要

摘要我们考虑有限域上的多项式f（x1，…，xn），它具有以下性质：对于域的某个元素b，方程f（x1…，xn）=b的解集与该域上某个线性方程组的解集一致。这样的多项式相对于右手边b是多仿射的。我们得到了有限域上多仿射多项式的性质。然后，我们证明了检查相对于给定右手边的多仿射可以通过具有多项式（根据变量的数量和输入多项式的和数）复杂性的算法来完成。除此之外，我们还证明了在正判决的情况下，可以恢复相应的线性方程组，其复杂性也是相同参数下的多项式。

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Multiaffine polynomials over a finite field

Abstract We consider polynomials f(x1, …, xn) over a finite field that possess the following property: for some element b of the field the set of solutions of the equation f(x1, …, xn) = b coincides with the set of solutions of some system of linear equations over this field. Such polynomials are said to be multiaffine with respect to the right-hand side b. We obtain the properties of multiaffine polynomials over a finite field. Then we show that checking the multiaffinity with respect to a given right-hand side may be done by an algorithm with polynomial (in terms of the number of variables and summands of the input polynomial) complexity. Besides that, we prove that in case of the positive decision a corresponding system of linear equations may be recovered with complexity which is also polynomial in terms of the same parameters.

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来源期刊

Discrete Mathematics and Applications MATHEMATICS, APPLIED-

CiteScore

0.60

自引率

20.00%

发文量

期刊介绍： The aim of this journal is to provide the latest information on the development of discrete mathematics in the former USSR to a world-wide readership. The journal will contain papers from the Russian-language journal Diskretnaya Matematika, the only journal of the Russian Academy of Sciences devoted to this field of mathematics. Discrete Mathematics and Applications will cover various subjects in the fields such as combinatorial analysis, graph theory, functional systems theory, cryptology, coding, probabilistic problems of discrete mathematics, algorithms and their complexity, combinatorial and computational problems of number theory and of algebra.