深度3恒等式的一个几乎最优秩界
An Almost Optimal Rank Bound for Depth-3 Identities
摘要：
我们证明了一个简单的、最小的和零的深度为3的回路（在任何域上）的秩至多为k^3 \log d。以前已知的最佳秩界是2^{O（k^2）}（\log d）^{k-2}，由Dvir和Shpilka（STOC 2005）提出。这几乎解决了Dvir和Shpilka首先提出的秩问题（因为我们还提供了秩Ω（k \log d）的简单和最小的恒等式）。我们的秩界显著改进了Karnin和Shpilka（CCC 2008）对深度3电路的最著名的确定性黑盒身份测试（对k的依赖性呈指数下降）。我们的技术还揭示了非零深度3电路的因式分解模式，最引人注目的是：一个简单的，最小的和非零深度3电路（在任何领域）的线性因子的秩最多是k^3 \log d。这项工作的新特点是一个新的概念之间的映射集的线性形式，称为“理想匹配”，用于研究深度3电路。我们证明有趣的结构结果深度3身份使用这些技术。我们相信，这些可以导致这些电路的确定性多项式时间身份测试的目标。
英文摘要：
We show that the rank of a depth-3 circuit (over any field) that is simple, minimal and zero is at most k^3\log d. The previous best rank bound known was 2^{O(k^2)}(\log d)^{k-2} by Dvir and Shpilka (STOC 2005). This almost resolves the rank question first posed by Dvir and Shpilka (as we also provide a simple and minimal identity of rank Ω(k\log d)). Our rank bound significantly improves (dependence on k exponentially reduced) the best known deterministic black-box identity tests for depth-3 circuits by Karnin and Shpilka (CCC 2008). Our techniques also shed light on the factorization pattern of nonzero depth-3 circuits, most strikingly: the rank of linear factors of a simple, minimal and nonzero depth-3 circuit (over any field) is at most k^3\log d. The novel feature of this work is a new notion of maps between sets of linear forms, called "ideal matchings", used to study depth-3 circuits. We prove interesting structural results about depth-3 identities using these techniques. We believe that these can lead to the goal of a deterministic polynomial time identity test for these circuits.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
