On Matrix Multiplication and Polynomial Identity Testing

IF 1.2 3区计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS

SIAM Journal on Computing Pub Date : 2024-02-27 DOI:10.1137/22m1536169

Robert Andrews

引用次数: 0

Abstract

SIAM Journal on Computing, Ahead of Print.
Abstract. We show that lower bounds on the border rank of matrix multiplication can be used to nontrivially derandomize polynomial identity testing for small algebraic circuits. Letting [math] denote the border rank of [math] matrix multiplication, we construct a hitting set generator with seed length [math] that hits [math]-variate circuits of multiplicative complexity [math]. If the matrix multiplication exponent [math] is not 2, our generator has seed length [math] and hits circuits of size [math] for sufficiently small [math]. Surprisingly, the fact that [math] already yields new, nontrivial hitting set generators for circuits of sublinear multiplicative complexity.

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关于矩阵乘法和多项式同一性检验

SIAM 计算期刊》，提前印刷。摘要我们证明，矩阵乘法边界秩的下限可用来对小型代数电路的多项式同一性检验进行非随机化。让[math]表示[math]矩阵乘法的边界秩，我们构建了一个种子长度为[math]的命中集生成器，它能命中乘法复杂度为[math]的[math]变量电路。如果矩阵乘法指数[math]不是 2，那么我们的生成器种子长度为[math]，在足够小的[math]条件下，能命中大小为[math]的电路。令人惊奇的是，[math]已经为具有亚线性乘法复杂度的电路提供了新的、非难的命中集生成器。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

SIAM Journal on Computing 工程技术-计算机：理论方法

CiteScore

4.60

自引率

0.00%

发文量

审稿时长

6-12 weeks

期刊介绍： The SIAM Journal on Computing aims to provide coverage of the most significant work going on in the mathematical and formal aspects of computer science and nonnumerical computing. Submissions must be clearly written and make a significant technical contribution. Topics include but are not limited to analysis and design of algorithms, algorithmic game theory, data structures, computational complexity, computational algebra, computational aspects of combinatorics and graph theory, computational biology, computational geometry, computational robotics, the mathematical aspects of programming languages, artificial intelligence, computational learning, databases, information retrieval, cryptography, networks, distributed computing, parallel algorithms, and computer architecture.