AI 中文总结
本文针对定义域为布尔切片、值域为一般阿贝尔群的低度测试问题，提出了查询复杂度为 $O_d(1)$ 的测试方法，通过将切片嵌入超立方体的归约思路完成分析，实现了对近距函数的高概率拒绝。
AI 中文摘要
我们研究布尔切片上群值函数的低度测试问题。具体而言，给定度数参数 $d$ 以及对函数 $f:\{0,1\}^n_{n/2}\to G$ 的预言访问，其中 $\{0,1\}^n_k$ 表示 $\{0,1\}^n$ 中汉明重量为 $k$ 的向量集合，$G$ 为阿贝尔群，低度测试问题要求我们区分两种情况：$f$ 是度数至多为 $d$（系数取自 $G$）的多项式，或者 $f$ 与所有此类多项式的集合的距离为 $\varepsilon$。该领域的经典工作考虑定义域为 $\mathbb{F}_q^n$、值域为 $\mathbb{F}_q$ 的函数；更近的工作则考虑定义域为布尔超立方体的情况[Bafna、Srinivasan、Sudan，《Random Struct. Algorithms》2020]，或定义域为切片（即 $\{0,1\}^n_k$）、值域为 $\mathbb{F}_2$ 的情况[David、Dinur、Goldenberg、Kindler 和 Shinkar，《SIAM J. Comput.》2017；Kalai、Lifshitz、Minzer 和 Ziegler，《FOCS》2024]。每一次定义域或值域的变更都会在低度测试的设计与分析中引入新的挑战，而在我们的研究场景中，定义域为切片、值域为一般群，这一情况再次带来了此类挑战。我们的主要定理给出了一个测试，该测试对 $f$ 进行 $O_d(1)$ 次查询，接受度数为 $d$ 的函数，同时以 $\Omega(\varepsilon)$ 的概率拒绝与该集合距离为 $\varepsilon$ 的函数。核心证明思路是将该低度测试问题归约为超立方体上的低度测试问题。具体而言，我们展示了如何将 $n/2$ 维超立方体 $\{0,1\}^{n/2}$ 随机嵌入 $n$ 维切片，同时近乎保持 $f$ 与该超立方体上度数为 $d$ 的多项式空间的接近度。尽管该嵌入简单自然，但其分析涉及细致的归纳，且新颖地使用了切片上度数为 $d$ 的多项式基（来自 Anstee、Rónyai 和 Sali 的《Graphs and Combinatorics》2002 年的工作）。
英文摘要
We study low-degree testing for group-valued functions over a Boolean slice. Specifically given a degree parameter $d$ and oracle access to a function $f:\{0,1\}^n_{n/2}\to G$ where $\{0,1\}^n_k$ denotes the set of vectors in $\{0,1\}^n$ of Hamming weight $k$ and $G$ is an Abelian group, the low-degree testing problem asks us to distinguish the case where $f$ is a polynomial of degree at most $d$ (with coefficients from $G$) or is $\varepsilon$-far from the set of all such polynomials. Classical works in this area considered functions with domain $\mathbb{F}_q^n$ and range $\mathbb{F}_q$. More recent works have considered the setting where the domain is the Boolean cube [Bafna, Srinivasan, Sudan (Random Struct. Algorithms 2020), Amireddy, Srinivasan, Sudan (RANDOM 2023)], or when the domain is the slice (i.e., $\{0,1\}^n_{k}$) and the range is $\mathbb{F}_2$ [David, Dinur, Goldenberg, Kindler and Shinkar (SIAM J. Comput. 2017), Kalai, Lifshitz, Minzer and Ziegler (FOCS 2024)]. Each of the changes introduces new challenges in designing and analyzing low-degree tests and this happens again in our setting with domain being a slice and range is general. Our main theorem gives a test that makes $O_d(1)$ queries to $f$ and accepts degree-$d$ functions while rejecting functions that are $\varepsilon$-far with probability $Ω(\varepsilon)$. The central proof idea is to reduce this low-degree testing problem to the problem of low-degree testing on the cube. Specifically we show how to randomly embed the $n/2$-dimensional cube $\{0,1\}^{n/2}$ in the $n$-dimensional slice while nearly preserving the proximity of $f$ to the space of degree-$d$ polynomials on this cube. While the embedding is simple and natural, the analysis involves a careful induction with a novel use of a basis of degree-$d$ polynomials on slices (from a work of Anstee, Rónyai and Sali (Graphs and Combinatorics 2002)).
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