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Hilbert Functions and Low-Degree Randomness Extractors

Alexander Golovnev, Zeyu Guo, Pooya Hatami, Satyajeet Nagargoje, Chao Yan

TL;DR

The bounds on the Hilbert function and degree-$d closure of sets are used to prove that a random low-degree polynomial is an extractor for samplable randomness sources and the existence of low-degree extractors and dispersers for sources generated by constant-degree polynomials and polynomial-size circuits is proved.

Abstract

For $S\subseteq \mathbb{F}^n$, consider the linear space of restrictions of degree-$d$ polynomials to $S$. The Hilbert function of $S$, denoted $\mathrm{h}_S(d,\mathbb{F})$, is the dimension of this space. We obtain a tight lower bound on the smallest value of the Hilbert function of subsets $S$ of arbitrary finite grids in $\mathbb{F}^n$ with a fixed size $|S|$. We achieve this by proving that this value coincides with a combinatorial quantity, namely the smallest number of low Hamming weight points in a down-closed set of size $|S|$. Understanding the smallest values of Hilbert functions is closely related to the study of degree-$d$ closure of sets, a notion introduced by Nie and Wang (Journal of Combinatorial Theory, Series A, 2015). We use bounds on the Hilbert function to obtain a tight bound on the size of degree-$d$ closures of subsets of $\mathbb{F}_q^n$, which answers a question posed by Doron, Ta-Shma, and Tell (Computational Complexity, 2022). We use the bounds on the Hilbert function and degree-$d$ closure of sets to prove that a random low-degree polynomial is an extractor for samplable randomness sources. Most notably, we prove the existence of low-degree extractors and dispersers for sources generated by constant-degree polynomials and polynomial-size circuits. Until recently, even the existence of arbitrary deterministic extractors for such sources was not known.

Hilbert Functions and Low-Degree Randomness Extractors

TL;DR

The bounds on the Hilbert function and degree-$d closure of sets are used to prove that a random low-degree polynomial is an extractor for samplable randomness sources and the existence of low-degree extractors and dispersers for sources generated by constant-degree polynomials and polynomial-size circuits is proved.

Abstract

For , consider the linear space of restrictions of degree- polynomials to . The Hilbert function of , denoted , is the dimension of this space. We obtain a tight lower bound on the smallest value of the Hilbert function of subsets of arbitrary finite grids in with a fixed size . We achieve this by proving that this value coincides with a combinatorial quantity, namely the smallest number of low Hamming weight points in a down-closed set of size . Understanding the smallest values of Hilbert functions is closely related to the study of degree- closure of sets, a notion introduced by Nie and Wang (Journal of Combinatorial Theory, Series A, 2015). We use bounds on the Hilbert function to obtain a tight bound on the size of degree- closures of subsets of , which answers a question posed by Doron, Ta-Shma, and Tell (Computational Complexity, 2022). We use the bounds on the Hilbert function and degree- closure of sets to prove that a random low-degree polynomial is an extractor for samplable randomness sources. Most notably, we prove the existence of low-degree extractors and dispersers for sources generated by constant-degree polynomials and polynomial-size circuits. Until recently, even the existence of arbitrary deterministic extractors for such sources was not known.
Paper Structure (44 sections, 59 theorems, 173 equations)

This paper contains 44 sections, 59 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Hilbert Functions.
  3. Randomness Extractors.
  4. Our Results on Hilbert Functions
  5. Degree-d Closure of Sets
  6. Our Results on Randomness Extractors
  7. Low-Degree Dispersers
  8. Low-Degree Extractors
  9. Uniform covering by sets of full Hilbert dimension.
  10. Remarks
  11. Correlation bounds over arbitrary subsets.
  12. Punctured Reed-Muller codes.
  13. Sampling lower bound for polynomial sources.
  14. Related Work.
  15. Acknowledgments.
  16. ...and 29 more sections

Key Result

Theorem 1.2

Let $\F$ be a field, and $A_1,\dots, A_n\subseteq \F$ be finite sets of size $|A_i|=r_i$. Define $A=A_1\times \cdots \times A_n$. For every $k\leq |A|$, where $F=\prod_i \{0,\dots, r_i-1\}$ and $T_{\leq d}=\{x\in T: \sum_i x_i \leq d\}$.

Theorems & Definitions (136)

  • Definition 1.1
  • Theorem 1.2: See \ref{['cor:characterization']}
  • Theorem 1.3: See \ref{['thm:hilbertbound']}
  • Corollary 1.4: See \ref{['cor:boundsOnHilbert']}
  • Theorem 1.5: nie2015hilbert
  • Theorem 1.6: See \ref{['thm:tight-bound-cl']} and \ref{['thm:closure-tightness']}
  • Corollary 1.7
  • proof
  • Example 1.8
  • Example 1.9
  • ...and 126 more