We study the question of local testability of low (constant) degree functions from a product domain $S_1 \times \dots \times {S}_n$ to a field $\mathbb{F}$, where ${S_i} \subseteq \mathbb{F}$ can be arbitrary constant sized sets. We show that this family is locally testable when the grid is "symmetric". That is, if ${S_i} = {S}$ for all i, there is a probabilistic algorithm using constantly many queries that distinguishes whether $f$ has a polynomial representation of degree at most $d$ or is $\Omega(1)$-far from having this property. In contrast, we show that there exist asymmetric grids with $|{S}_1| =\dots= |{S}_n| = 3$ for which testing requires $\omega_n(1)$ queries, thereby establishing that even in the context of polynomials, local testing depends on the structure of the domain and not just the distance of the underlying code. The low-degree testing problem has been studied extensively over the years and a wide variety of tools have been applied to propose and analyze tests. Our work introduces yet another new connection in this rich field, by building low-degree tests out of tests for "junta-degrees". A function $f : {S}_1 \times \dots \times {S}_n \to {G}$, for an abelian group ${G}$ is said to be a junta-degree-$d$ function if it is a sum of $d$-juntas. We derive our low-degree test by giving a new local test for junta-degree-$d$ functions. For the analysis of our tests, we deduce a small-set expansion theorem for spherical noise over large grids, which may be of independent interest.

翻译：我们研究从乘积域 $S_1 \times \dots \times {S}_n$ 到域 $\mathbb{F}$ 的低次（常数次）函数的局部可测试性问题，其中 ${S_i} \subseteq \mathbb{F}$ 可以是任意常数大小的集合。我们证明，当网格具有“对称性”时，这一函数族是局部可测试的。也就是说，如果对所有 $i$ 都有 ${S_i} = {S}$，则存在一个使用常数次查询的概率算法，能够区分函数 $f$ 是否具有次数至多为 $d$ 的多项式表示，或者与该性质的距离是否为 $\Omega(1)$。相反，我们证明了存在满足 $|{S}_1| =\dots= |{S}_n| = 3$ 的非对称网格，其测试需要 $\omega_n(1)$ 次查询，从而表明即使在多项式环境中，局部测试也依赖于域的结构，而不仅仅是底层码的距离。低次测试问题多年来已被广泛研究，人们应用了多种工具来提出和分析测试方法。我们的工作通过从“轮辐次数”（junta-degree）测试中构建低次测试，为这一丰富领域引入了另一种新的联系。如果函数 $f : {S}_1 \times \dots \times {S}_n \to {G}$（其中 ${G}$ 为阿贝尔群）可以表示为 $d$-轮辐函数之和，则称其为轮辐次数-$d$ 函数。我们通过为轮辐次数-$d$ 函数给出一种新的局部测试，推导出我们的低次测试。在分析我们的测试时，我们推导出了一个关于大网格上球形噪声的小集扩张定理，该定理可能具有独立的意义。