Algorithmizing the Multiplicity Schwartz-Zippel Lemma
Siddharth Bhandari, Prahladh Harsha, Mrinal Kumar, Ashutosh Shankar · Society for Industrial and Applied Mathematics eBooks · 2023
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 numerous 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 arbitrary 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 underlying 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 [BHKS21b]. 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 algorithm 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 analysis of Forney's classical generalized minimum distance decoder that might be of independent interest. * The full version of the paper which includes the missing proofs can be accessed at [BHKS21a]. Research of the first, second and fourth authors supported by the Department of Atomic Energy, Government of India, under project 12-R&D-TFR-5.01-0500. This work was done while the first author was at TIFR, where he was supported in part by the Google PhD Fellowship and at the Simons Institute for the Theory of Computing where he was supported by the Simons-Berkeley Postdoctoral Fellowship. Research of the second author supported in part by the Swarnajayanti Fellowship.