The well-known DeMillo-Lipton-Schwartz-Zippel lemma says that $n$-variate
polynomials of total degree at most $d$ over
grids, i.e. sets of the form $A_1 \times A_2 \times \cdots \times A_n$, form
error-correcting codes (of distance at least $2^{-d}$ provided $\min_i\{|A_i|\}\geq 2$).
In this work we explore their local
decodability and (tolerant) local testability. While these aspects have been studied
extensively when $A_1 = \cdots = A_n = \F_q$ are the same finite field, the
setting when $A_i$'s are not the full field does not seem to have been explored before.

In this work we focus on the case $A_i = \{0,1\}$ for every $i$. We show that for every field
(finite or otherwise) there is a test whose query complexity depends only on the
degree (and not on the number of variables). In contrast we show that
decodability is possible over fields of positive characteristic (with query complexity growing with
the degree of the polynomial and the characteristic), but not over the reals, where the query complexity must grow with $n$. As a
consequence we get a natural example of a code (one with a transitive group of symmetries) that is locally testable
but not locally decodable.

Classical results on local decoding and testing of polynomials have relied on the
2-transitive symmetries of the space of low-degree polynomials (under affine
transformations). Grids do not possess this symmetry: So we introduce some new
techniques to overcome this handicap and in particular use the hypercontractivity of
the (constant weight) noise operator on the Hamming cube.

Changed author list. Added new results on tolerant testing (Section 7).

The well-known DeMillo-Lipton-Schwartz-Zippel lemma says that $n$-variate
polynomials of total degree at most $d$ over
grids, i.e. sets of the form $A_1 \times A_2 \times \cdots \times A_n$, form
error-correcting codes (of distance at least $2^{-d}$ provided $\min_i\{|A_i|\}\geq 2$).
In this work we explore their local
decodability and local testability. While these aspects have been studied
extensively when $A_1 = \cdots = A_n = \F_q$ are the same finite field, the
setting when $A_i$'s are not the full field does not seem to have been explored before.

In this work we focus on the case $A_i = \{0,1\}$ for every $i$. We show that for every field
(finite or otherwise) there is a test whose query complexity depends only on the
degree (and not on the number of variables). In contrast we show that
decodability is possible over fields of positive characteristic (with query complexity growing with
the degree of the polynomial and the characteristic), but not over the reals, where the query complexity must grow with $n$. As a
consequence we get a natural example of a code (one with a transitive group of symmetries) that is locally testable
but not locally decodable.

Classical results on local decoding and testing of polynomials have relied on the
2-transitive symmetries of the space of low-degree polynomials (under affine
transformations). Grids do not possess this symmetry: So we introduce some new
techniques to overcome this handicap and in particular use the hypercontractivity of
the (constant weight) noise operator on the Hamming cube.