Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Polynomial Identity Testing via Evaluation of Rational Functions

Ivan Hu, Dieter van Melkebeek, Andrew Morgan

TL;DR

This work studies polynomial identity testing through hitting-set generators built from evaluations of low-degree univariate rational functions at abscissas tied to the variables. It shows the Rational Function Evaluation (RFE) generator is equivalent, up to rescaling, to the Shpilka–Volkovich (SV) generator and develops a rigorous vanishing-ideal framework to analyze their power. The authors give two structural characterizations of the vanishing ideal: a small, explicit generating set via Elementary Vandermonde Circulations (EVC) and a deterministic, multilinear membership test based on zero substitutions and partial derivatives, plus a variant for set-multilinearity with Elementary Set-Multilinear Vandermonde Circulations (ESMVC). They rederive known PIT derandomization results, extend them to read-once formulas and ROABPs, and provide new lower-bound insights using the vanishing-ideal perspective. The paper also connects to alternating algebra to illuminate the underlying combinatorial and algebraic structure, suggesting broader implications for derandomization and hardness in algebraic computation.

Abstract

$ \newcommand{\ie}{i.\,e.} $We introduce a hitting set generator for Polynomial Identity Testing based on evaluations of low-degree univariate rational functions at abscissas associated with the variables. We establish an equivalence up to rescaling with a generator introduced by Shpilka and Volkovich, which has a similar structure but uses multivariate polynomials. We initiate a systematic analytic study of the power of hitting set generators by characterizing their vanishing ideals, \ie, the sets of polynomials that they fail to hit. We provide two such characterizations for our generator. First, we develop a small collection of polynomials that jointly produce the vanishing ideal. As corollaries, we obtain tight bounds on the minimum degree, sparseness, and partition class size of set-multilinearity in the vanishing ideal. Second, inspired by a connection to alternating algebra, we develop a structured deterministic membership test for the multilinear part of the vanishing ideal. We present a derivation based on alternating algebra along with the required background, as well as one in terms of zero substitutions and partial derivatives, avoiding the need for alternating algebra. As evidence of the utility of our analytic approach, we rederive known derandomization results based on the generator by Shpilka and Volkovich and present a new application in derandomization / lower bounds for read-once oblivious algebraic branching programs.

Polynomial Identity Testing via Evaluation of Rational Functions

TL;DR

This work studies polynomial identity testing through hitting-set generators built from evaluations of low-degree univariate rational functions at abscissas tied to the variables. It shows the Rational Function Evaluation (RFE) generator is equivalent, up to rescaling, to the Shpilka–Volkovich (SV) generator and develops a rigorous vanishing-ideal framework to analyze their power. The authors give two structural characterizations of the vanishing ideal: a small, explicit generating set via Elementary Vandermonde Circulations (EVC) and a deterministic, multilinear membership test based on zero substitutions and partial derivatives, plus a variant for set-multilinearity with Elementary Set-Multilinear Vandermonde Circulations (ESMVC). They rederive known PIT derandomization results, extend them to read-once formulas and ROABPs, and provide new lower-bound insights using the vanishing-ideal perspective. The paper also connects to alternating algebra to illuminate the underlying combinatorial and algebraic structure, suggesting broader implications for derandomization and hardness in algebraic computation.

Abstract

We introduce a hitting set generator for Polynomial Identity Testing based on evaluations of low-degree univariate rational functions at abscissas associated with the variables. We establish an equivalence up to rescaling with a generator introduced by Shpilka and Volkovich, which has a similar structure but uses multivariate polynomials. We initiate a systematic analytic study of the power of hitting set generators by characterizing their vanishing ideals, \ie, the sets of polynomials that they fail to hit. We provide two such characterizations for our generator. First, we develop a small collection of polynomials that jointly produce the vanishing ideal. As corollaries, we obtain tight bounds on the minimum degree, sparseness, and partition class size of set-multilinearity in the vanishing ideal. Second, inspired by a connection to alternating algebra, we develop a structured deterministic membership test for the multilinear part of the vanishing ideal. We present a derivation based on alternating algebra along with the required background, as well as one in terms of zero substitutions and partial derivatives, avoiding the need for alternating algebra. As evidence of the utility of our analytic approach, we rederive known derandomization results based on the generator by Shpilka and Volkovich and present a new application in derandomization / lower bounds for read-once oblivious algebraic branching programs.
Paper Structure (37 sections, 50 theorems, 109 equations, 1 figure)

This paper contains 37 sections, 50 theorems, 109 equations, 1 figure.

Table of Contents

  1. Overview
  2. Vanishing ideal.
  3. Rational function evaluations.
  4. Generating set.
  5. Membership test.
  6. Applications.
  7. Derandomization.
  8. Lower bounds.
  9. Techniques.
  10. Alternating algebra representation.
  11. Related recent work and further research.
  12. Organization.
  13. RFE Generator
  14. Generating Set
  15. Gröbner basis.
  16. ...and 22 more sections

Key Result

theorem 1

Let $k, l, n \in \N$ and let $a_i$ for $i \in [n]$ be distinct elements of $\mathbb{F}$. The vanishing ideal of $\mathrm{RFE}^{k}_{l}$ in $\mathbb{F}[x_1,\dots,x_n]$ for the given choice of abscissas $(a_i)_{i\in[n]}$ is generated by the following polynomials over all choices of $k+l+2$ indices $i_1 Moreover, for any fixed set $C \subseteq [n]$ of size $k+1$, the polynomials ${{\mathrm{EVC}}^{k}_{

Figures (1)

  • Figure 1: Rank lower bound analysis in terms of the blocks $C_d$ of $p$ and $C'_d$ of $p'$ (Proposition \ref{['prop:ROABP:zero']})

Theorems & Definitions (107)

  • definition 1: SV generator
  • definition 2: RFE generator
  • theorem 1: generating set
  • corollary 1
  • corollary 2
  • corollary 3
  • theorem 2: membership test for multilinear polynomials
  • theorem 3: ROABP hitting property
  • theorem 4: ROABP lower bound
  • lemma 1
  • ...and 97 more