具有小偏导数空间的多项式的命中集

中文摘要 AI 辅助

我们在特征为零的任意域 $\mathbb{F}$ 上，为偏导数空间以 $r$ 为界的 $n$ 元 $d$ 次多项式类，给出了一个大小为 $\text{poly}(n,d,r)$ 的显式命中集。特别地，这为深度为 $3$ 的幂电路类生成了一个多项式大小的命中集。主要的技术洞见是构造了一个“形式导数”以及与该导数相关的 Wronskian 的性质，这一性质此前由 Moura [Moura_2004] 在完全不同的背景下研究过。本文中的证明是初等的且完全自包含的。AI 披露：该结果的证明是在与 OpenAI GPT-6 Astra 的对话 [astra_proof] 中获得的。本文所呈现的证明是对 AI 模型所获得证明的重写（以作者自己的话），我们相信这种形式对研究人员来说是可理解的。

英文摘要

We give an explicit hitting set of size $\text{poly}(n,d,r)$ for the class of $n$-variate degree-$d$ polynomials whose partial derivative space is bounded by $r$, over any field $\mathbb{F}$ of characteristic zero. In particular, this yields a polynomial sized hitting set for the class of depth-$3$ powering circuits. The main technical insight is the construction of a "formal derivation'' and properties of the associated Wronskian with respect to this derivation, which was previously studied by Moura [Moura_2004] in a very different context. The proofs in this paper are elementary and completely self-contained. AI disclosure: The proof of this result was obtained during conversations [astra_proof] with OpenAI GPT-6 Astra. The proof presented in this writeup is a rewriting (in the authors' words) of the proof obtained by the AI model in a form that we believe is understandable to researchers.

发表机构

• Tata Institute of Fundamental Research（塔塔基础研究所）

机构由 AI 辅助整理，请以论文原文为准。