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Algorithmizing the Multiplicity Schwartz-Zippel Lemma

Siddharth Bhandari, Prahladh Harsha, Mrinal Kumar, Ashutosh Shankar

TL;DR

The paper delivers an algorithmic version of the multiplicity Schwartz-Zippel lemma for arbitrary product sets over any field, enabling efficient unique decoding of multivariate multiplicity codes from half their minimum distance in the constant-multiplicity regime. It extends Kim-Kopparty’s Reed-Muller decoding framework to s>1 by introducing a refined, multiplicity-aware distance and a suite of univariate and weighted decoders, subsequently lifting the approach to the multivariate setting (including m>2) through an alternative SZ-based analysis of decoding. A central contribution is the weighted univariate multiplicity code decoder and its integration into a bivariate core, with a detailed generalization to multivariate cases, plus an alternative Forney GMD analysis to underpin the thresholding steps. The results yield a concrete, parameter-dependent decoding algorithm over arbitrary grids, with running time polynomial in the input size for constant s and practical for coding-theoretic applications, while leaving open questions about polynomial-time performance in all regimes and extensions beyond half-distance.

Abstract

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 [SIAM J. Comput., 2013], 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 [J. ACM, 2014]. 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 (by Kopparty [ToC, 2015]) 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 $1$ (Bhandari, Harsha, Kumar and Sudan [STOC 2021]). 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 and Kopparty [ToC, 2017] 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.

Algorithmizing the Multiplicity Schwartz-Zippel Lemma

TL;DR

The paper delivers an algorithmic version of the multiplicity Schwartz-Zippel lemma for arbitrary product sets over any field, enabling efficient unique decoding of multivariate multiplicity codes from half their minimum distance in the constant-multiplicity regime. It extends Kim-Kopparty’s Reed-Muller decoding framework to s>1 by introducing a refined, multiplicity-aware distance and a suite of univariate and weighted decoders, subsequently lifting the approach to the multivariate setting (including m>2) through an alternative SZ-based analysis of decoding. A central contribution is the weighted univariate multiplicity code decoder and its integration into a bivariate core, with a detailed generalization to multivariate cases, plus an alternative Forney GMD analysis to underpin the thresholding steps. The results yield a concrete, parameter-dependent decoding algorithm over arbitrary grids, with running time polynomial in the input size for constant s and practical for coding-theoretic applications, while leaving open questions about polynomial-time performance in all regimes and extensions beyond half-distance.

Abstract

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 [SIAM J. Comput., 2013], 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 [J. ACM, 2014]. 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 (by Kopparty [ToC, 2015]) 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 (Bhandari, Harsha, Kumar and Sudan [STOC 2021]). 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 and Kopparty [ToC, 2017] 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.
Paper Structure (36 sections, 15 theorems, 47 equations, 6 figures, 4 algorithms)

This paper contains 36 sections, 15 theorems, 47 equations, 6 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. The coding theoretic context and main theorem
  3. Alternative analysis of Forney's GMD Decoding
  4. Further discussion and open problems
  5. Organization
  6. Overview of algorithm
  7. An overview of the Kim-Kopparty Reed-Muller decoder
  8. Generalizing the Kim-Kopparty decoder to s > 1
  9. Extracting P l from the fractional word
  10. The multivariate setting for m > 2
  11. Multiplicity codes and main result
  12. Notation
  13. Multiplicity code
  14. A fine-grained notion of distance for multiplicity codewords
  15. Main Result
  16. ...and 21 more sections

Key Result

Lemma 3.5

Let $P \in \mathbb{F}[\mathbf{x}]$ be a nonzero $m$-variate polynomial of total degree at most $d$ and let $T \subseteq \mathbb{F}$. Then,

Figures (6)

  • Figure 4: Bivariate algorithm, visualised
  • Figure 5: Step-threshold in weighted univariate multiplicity decoding
  • Figure 6: Encoding of a concatenated code
  • Figure 7: Inner decoding of a concatenated code
  • Figure 9: When $a$ and $b$ can be in a good pairing, with $\omega(b) \leq \omega(a)$
  • ...and 1 more figures

Theorems & Definitions (33)

  • remark 1.1
  • definition 3.1: Multiplicity code
  • remark 3.2
  • remark 3.3
  • definition 3.4: multiplicity at a point
  • Lemma 3.5: multiplicity Schwartz-Zippel lemma DvirKSS2013
  • definition 3.6
  • remark 3.7
  • Theorem 3.9
  • claim 4.1
  • ...and 23 more