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A weak regularity lemma for polynomials

Guy Moshkovitz, Dora Woodruff

TL;DR

This work presents a weak regularity lemma for polynomials over finite fields, delivering strong, non-tower bounds that enable quantitative analysis of curves in the image of a polynomial map. The authors develop a rank-regularity framework and a bias-based Fourier approach to derive a weak regular decomposition of size $(2m(r+1))^{2^{d(1+o(1))}}$ for degree-$d$ polynomial arrays, with the key consequence that high univariate degree in the image forces small rank parameters and bounded top fan-in in depth-4 arithmetic formulas. A central application is a bound on $\\mathop{rank}_{d/2}(\\mathbf{P})$ (and more generally $\\mathop{rank}_{d/udeg(\\mathbf{P})}$) for polynomial maps whose image contains no line, linking algebraic structure to circuit complexity and revealing a barrier to certain arithmetic lower-bound techniques. The results illuminate the role of univariate degree as a critical parameter in understanding polynomial maps and arithmetic circuits, and they provide new tools to analyze the geometry of polynomial images and their implications for complexity theory.

Abstract

A regularity lemma for polynomials provides a decomposition in terms of a bounded number of approximately independent polynomials. Such regularity lemmas play an important role in numerous results, yet suffer from the familiar shortcoming of having tower-type bounds or worse. In this paper we design a new, weaker regularity lemma with strong bounds. The new regularity lemma in particular provides means to quantitatively study the curves contained in the image of a polynomial map, which is beyond the reach of standard methods. Applications include strong bounds for a problem of Karam on generalized rank, as well as a new method to obtain upper bounds for fan-in parameters in arithmetic circuits. For example, we show that if the image of a polynomial map $\mathbf{P} \colon \mathbb{F}^n \to \mathbb{F}^m$ of degree $d$ does not contain a line, then $\mathbf{P}$ can be computed by a depth-$4$ arithmetic formula with bottom fan-in at most $d/2$ and top fan-in at most $(2m)^{C(d)}$ (with $C(d)=2^{(1+o(1))d}$). One implication of our work is a certain ``barrier'' to arithmetic circuit lower bounds, in terms of the smallest degree of a polynomial curve contained in the image of the given polynomial map.

A weak regularity lemma for polynomials

TL;DR

This work presents a weak regularity lemma for polynomials over finite fields, delivering strong, non-tower bounds that enable quantitative analysis of curves in the image of a polynomial map. The authors develop a rank-regularity framework and a bias-based Fourier approach to derive a weak regular decomposition of size for degree- polynomial arrays, with the key consequence that high univariate degree in the image forces small rank parameters and bounded top fan-in in depth-4 arithmetic formulas. A central application is a bound on (and more generally ) for polynomial maps whose image contains no line, linking algebraic structure to circuit complexity and revealing a barrier to certain arithmetic lower-bound techniques. The results illuminate the role of univariate degree as a critical parameter in understanding polynomial maps and arithmetic circuits, and they provide new tools to analyze the geometry of polynomial images and their implications for complexity theory.

Abstract

A regularity lemma for polynomials provides a decomposition in terms of a bounded number of approximately independent polynomials. Such regularity lemmas play an important role in numerous results, yet suffer from the familiar shortcoming of having tower-type bounds or worse. In this paper we design a new, weaker regularity lemma with strong bounds. The new regularity lemma in particular provides means to quantitatively study the curves contained in the image of a polynomial map, which is beyond the reach of standard methods. Applications include strong bounds for a problem of Karam on generalized rank, as well as a new method to obtain upper bounds for fan-in parameters in arithmetic circuits. For example, we show that if the image of a polynomial map of degree does not contain a line, then can be computed by a depth- arithmetic formula with bottom fan-in at most and top fan-in at most (with ). One implication of our work is a certain ``barrier'' to arithmetic circuit lower bounds, in terms of the smallest degree of a polynomial curve contained in the image of the given polynomial map.

Paper Structure

This paper contains 16 sections, 21 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Weak regularity for polynomials
  3. Applications
  4. The proof
  5. Organization
  6. Preliminaries
  7. Acknowledgments:
  8. Weak Regularity
  9. Rank-regularity
  10. Linear spaces of polynomials
  11. Proof of the rank-regularity lemma
  12. Bias and weak regularity
  13. Putting everything together
  14. Applications
  15. Properties of univariate degree
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\epsilon=q^{-r}$ with $r > 1$. Every polynomial $m$-tuple $\mathbf{P}$ over $\mathbb{F}=\mathbb{F}_q$, of degrees at most $d$ with $d < \mathop{\mathrm{char}}\nolimits(\mathbb{F})$, has a weak $\epsilon$-regular decomposition of size at most $(mr)^{2^{d(1+o(1))}}$.

Theorems & Definitions (45)

  • Definition : weak regularity
  • Theorem : weak regularity lemma
  • Theorem : see Theorem \ref{['thm:surj']}
  • Theorem 1.1: Karam, Theorem 1.3 in Karam23
  • Theorem : see Theorem \ref{['thm:rk-bd']}
  • Theorem 1.2
  • Remark 1.3
  • Definition 2.1: weak $\epsilon$-regularity
  • Theorem 2.2: weak regularity lemma
  • Definition 2.3: rank
  • ...and 35 more