We give a polynomial-time algorithm to decode multivariate polynomial codes of degree $d$ up to half their minimum distance, when the evaluation points are an arbitrary product set $S^m$, for every $d < |S|$. Previously known algorithms could achieve this only if the set $S$ had some very special algebraic structure, or if the degree $d$ was significantly smaller than $|S|$. We also give a near-linear-time randomized algorithm, based on tools from list-decoding, to decode these codes from nearly half their minimum distance, provided $d < (1-\epsilon)|S|$ for constant $\epsilon > 0$. Our result gives an $m$-dimensional generalization of the well-known decoding algorithms for Reed--Solomon codes, and can be viewed as giving an algorithmic version of the Schwartz--Zippel lemma.

A conference version of this paper appeared in the Proceedings of the 31st Conference on Computational Complexity, 2016.