The multiplicity Schwartz-Zippel lemma asserts that over a field, a low-degree polynomial cannot vanish with high multiplicity very often on a sufficiently large product set. Since its discovery in a work of Dvir, Kopparty, Saraf and Sudan [DKSS13], the lemma has found nu- merous applications in both math and computer science; in particular, in the definition and properties of multiplicity codes by Kopparty, Saraf and Yekhanin [KSY14].

In this work, we show how to algorithmize the multiplicity Schwartz-Zippel lemma for ar- bitrary product sets over any field. In other words, we give an efficient algorithm for unique decoding of multivariate multiplicity codes from half their minimum distance on arbitrary product sets over all fields. Previously, such an algorithm was known either when the un- derlying product set had a nice algebraic structure (for instance, was a subfield) [Kop15] or when the underlying field had large (or zero) characteristic, the multiplicity parameter was sufficiently large and the multiplicity code had distance bounded away from 1 [BHKS21]. In particular, even unique decoding of bivariate multiplicity codes with multiplicity two from half their minimum distance was not known over arbitrary product sets over any field.

Our algorithm builds upon a result of Kim & Kopparty [KK17] who gave an algorithmic version of the Schwartz-Zippel lemma (without multiplicities) or equivalently, an efficient al- gorithm for unique decoding of Reed-Muller codes over arbitrary product sets. We introduce a refined notion of distance based on the multiplicity Schwartz-Zippel lemma and design a unique decoding algorithm for this distance measure. On the way, we give an alternate proof of Forney’s classical generalized minimum distance decoder that might be of independent interest.

Fixed typos and elaborated some proofs

The multiplicity Schwartz-Zippel lemma asserts that over a field, a low-degree polynomial cannot vanish with high multiplicity very often on a sufficiently large product set. Since its discovery in a work of Dvir, Kopparty, Saraf and Sudan [DKSS13], the lemma has found nu- merous applications in both math and computer science; in particular, in the definition and properties of multiplicity codes by Kopparty, Saraf and Yekhanin [KSY14].

In this work, we show how to algorithmize the multiplicity Schwartz-Zippel lemma for ar- bitrary product sets over any field. In other words, we give an efficient algorithm for unique decoding of multivariate multiplicity codes from half their minimum distance on arbitrary product sets over all fields. Previously, such an algorithm was known either when the un- derlying product set had a nice algebraic structure (for instance, was a subfield) [Kop15] or when the underlying field had large (or zero) characteristic, the multiplicity parameter was sufficiently large and the multiplicity code had distance bounded away from 1 [BHKS21]. In particular, even unique decoding of bivariate multiplicity codes with multiplicity two from half their minimum distance was not known over arbitrary product sets over any field.

Our algorithm builds upon a result of Kim & Kopparty [KK17] who gave an algorithmic version of the Schwartz-Zippel lemma (without multiplicities) or equivalently, an efficient al- gorithm for unique decoding of Reed-Muller codes over arbitrary product sets. We introduce a refined notion of distance based on the multiplicity Schwartz-Zippel lemma and design a unique decoding algorithm for this distance measure. On the way, we give an alternate proof of Forney’s classical generalized minimum distance decoder that might be of independent interest.