Decoding Reed-Muller Codes over Product Sets

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 $d 0$. 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.