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：我们研究了从积域$S_1 \times \dots \times {S}_n$到域$\mathbb{F}$的低（常数）阶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"对称"时，该函数族是局部可测试的。也就是说，若对所有i满足${S_i} = {S}$，则存在一种使用常数数量查询的概率算法，能够区分函数$f$是否具有至多$d$阶的多项式表示，或是$\Omega(1)$远距具备该性质。相比之下，我们证明存在满足$|{S}_1| =\dots= |{S}_n| = 3$的非对称网格，其测试需要$\omega_n(1)$次查询，从而确立即使在多项式语境下，局部测试也取决于域的结构而不仅仅是底层编码的距离。低阶测试问题多年来已得到广泛研究，多种工具被用于提出和分析测试方案。我们的工作通过基于"junta-degree"测试构建低阶测试，为这一丰富领域引入了新的联系。对于阿贝尔群${G}$，若函数$f : {S}_1 \times \dots \times {S}_n \to {G}$可表示为$d$-juntas的和，则称其为junta-degree-$d$函数。我们通过为junta-degree-$d$函数提供新的局部测试，推导出低阶测试方案。在测试分析中，我们推导了大网格上球面噪声的小集合扩展定理，该定理可能具有独立的研究价值。