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Feasibly Constructive Proof of Schwartz-Zippel Lemma and the Complexity of Finding Hitting Sets

Albert Atserias, Iddo Tzameret

Abstract

The Schwartz-Zippel Lemma states that if a low-degree multivariate polynomial with coefficients in a field is not zero everywhere in the field, then it has few roots on every finite subcube of the field. This fundamental fact about multivariate polynomials has found many applications in algorithms, complexity theory, coding theory, and combinatorics. We give a new proof of the lemma that offers some advantages over the standard proof. First, the new proof is more constructive than previously known proofs. For every given side-length of the cube, the proof constructs a polynomial-time computable and polynomial-time invertible surjection onto the set of roots in the cube. The domain of the surjection is tight, thus showing that the set of roots on the cube can be compressed. Second, the new proof can be formalised in Buss' bounded arithmetic theory $\mathrm{S}^1_2$ for polynomial-time reasoning. One consequence of this is that the theory $\mathrm{S}^1_2 + \mathrm{dWPHP(PV)}$ for approximate counting can prove that the problem of verifying polynomial identities (PIT) can be solved by polynomial-size circuits. The same theory can also prove the existence of small hitting sets for any explicitly described class of polynomials of polynomial degree. To complete the picture we show that the existence of such hitting sets is \emph{equivalent} to the surjective weak pigeonhole principle $\mathrm{dWPHP(PV)}$, over the theory $\mathrm{S}^1_2$. This is a contribution to a line of research studying the reverse mathematics of computational complexity. One consequence of this is that the problem of constructing small hitting sets for such classes is complete for the class APEPP of explicit construction problems whose totality follows from the probabilistic method. This class is also known and studied as the class of Range Avoidance Problems.

Feasibly Constructive Proof of Schwartz-Zippel Lemma and the Complexity of Finding Hitting Sets

Abstract

The Schwartz-Zippel Lemma states that if a low-degree multivariate polynomial with coefficients in a field is not zero everywhere in the field, then it has few roots on every finite subcube of the field. This fundamental fact about multivariate polynomials has found many applications in algorithms, complexity theory, coding theory, and combinatorics. We give a new proof of the lemma that offers some advantages over the standard proof. First, the new proof is more constructive than previously known proofs. For every given side-length of the cube, the proof constructs a polynomial-time computable and polynomial-time invertible surjection onto the set of roots in the cube. The domain of the surjection is tight, thus showing that the set of roots on the cube can be compressed. Second, the new proof can be formalised in Buss' bounded arithmetic theory for polynomial-time reasoning. One consequence of this is that the theory for approximate counting can prove that the problem of verifying polynomial identities (PIT) can be solved by polynomial-size circuits. The same theory can also prove the existence of small hitting sets for any explicitly described class of polynomials of polynomial degree. To complete the picture we show that the existence of such hitting sets is \emph{equivalent} to the surjective weak pigeonhole principle , over the theory . This is a contribution to a line of research studying the reverse mathematics of computational complexity. One consequence of this is that the problem of constructing small hitting sets for such classes is complete for the class APEPP of explicit construction problems whose totality follows from the probabilistic method. This class is also known and studied as the class of Range Avoidance Problems.

Paper Structure

This paper contains 42 sections, 20 theorems, 52 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Quest for Feasibly Constructive Proofs
  3. New Proof of the Schwartz-Zippel Lemma
  4. Applications
  5. Schwartz-Zippel Lemma in the Theory
  6. Existence of Hitting Sets and PIT in the Theory
  7. Contribution to Reverse Mathematics of Complexity Theory
  8. Application to Computational Complexity and Range Avoidance Problem
  9. Preliminaries
  10. Theories of Bounded Arithmetic
  11. The language $\mathcal{L}$.
  12. The Theory ${\mathsf{S^1_2}}({\mathsf{PV}})$.
  13. Definable sets and set-bounded quantification.
  14. Dual Weak Pigeonhole Principle.
  15. Polynomials and Algebraic Circuits
  16. ...and 27 more sections

Key Result

Theorem 1.1

Let $\mathbb{F}$ be a field, let $\overline x$ be a set of $n$ indeterminates, let $S \subseteq \mathbb{F}$ be a finite subset of field elements, and let $P(\overline x)$ be a multivariate polynomial in the indeterminates $\overline x$ with coefficients in the field $\mathbb{F}$ and maximum individu

Figures (1)

  • Figure :

Theorems & Definitions (45)

  • Theorem 1.1: Schwartz-Zippel Lemma
  • Theorem 1.2: Schwartz-Zippel Lemma in $\mathsf{S^1_2}$; informal, see \ref{['lem:encoding-roots']}
  • Theorem 1.3: Half of Fundamental Theorem of Algebra in $\mathsf{S^1_2}$ informal; see Lemma \ref{['lem:univariate']}
  • Theorem 1.4: Small Hitting Sets Exist in ${\mathsf{S^1_2}}+\mathsf{dWPHP}({\mathsf{PV}})$; informal, see \ref{['lem:hittingset']}
  • Theorem 1.5: Reverse Mathematics of Hitting Sets; informal, see Theorem \ref{['thm:equivalence']}
  • Theorem 1.6: Completeness of Finding Hitting Sets; informal, see Theorem \ref{['thm:intro:completeness']}
  • Definition 3.1: In ${\mathsf{S^1_2}}$
  • Theorem 3.2: Fundamental Theorem of Algebra
  • Remark 3.3
  • Lemma 3.4: Second Half of Fundamental Theorem of Algebra in ${\mathsf{S^1_2}}$
  • ...and 35 more