  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.
