矩阵刚性与Croot-Lev-Pach引理
Matrix rigidity and the Croot-Lev-Pach lemma
摘要：
矩阵刚性是Valiant提出的一个概念，作为证明算术电路下界的一种手段。一个矩阵是刚性的，如果它在汉明距离上远离任何低秩矩阵。尽管几十年的努力，没有明确的矩阵刚性足以执行Valiant的计划已经found. Recently，阿尔曼和威廉姆斯表明，相反的共同信念，$2^n \times 2^n$阿达玛矩阵不能用于Valiant的程序，因为它是不够刚性。在本文中，我们观察到一个类似的“非刚性”的现象，任何$q^n \times q^n$矩阵$M$的形式$M(x,y) = f(x+y)$，其中$f:F_q^n \to F_q$是任何功能和$F_q$是一个固定的有限域的$q$元素（$n$走向无穷大）。该定理遵循几乎立即从最近引理的Croot，列夫和Pach这也是主要成分在最近的解决方案的帽集问题。
英文摘要：
Matrix rigidity is a notion put forth by Valiant as a means for proving arithmetic circuit lower bounds. A matrix is rigid if it is far, in Hamming distance, from any low rank matrix. Despite decades of efforts, no explicit matrix rigid enough to carry out Valiant's plan has been found. Recently, Alman and Williams showed, contrary to common belief, that the $2^n \times 2^n$ Hadamard matrix could not be used for Valiant's program as it is not sufficiently rigid. In this note we observe a similar `non rigidity' phenomena for any $q^n \times q^n$ matrix $M$ of the form $M(x,y) = f(x+y)$, where $f:F_q^n \to F_q$ is any function and $F_q$ is a fixed finite field of $q$ elements ($n$ goes to infinity). The theorem follows almost immediately from a recent lemma of Croot, Lev and Pach which is also the main ingredient in the recent solution of the cap-set problem.
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