Extractors for Images of Varieties

TL;DR

This paper develops explicit deterministic extractors for polynomial images of varieties over finite fields, generalizing previous sources and working over large fields, with applications including Noether normalization and affine extractors.

Contribution

It introduces a new construction of deterministic extractors for polynomial images of varieties, extending prior work to larger fields and more general sources.

Findings

Extractor works over large finite fields of arbitrary characteristic.

Provides explicit Noether normalization lemmas for affine varieties.

Generalizes affine extractors to all finite prime fields of quasipolynomial size.

Abstract

We construct explicit deterministic extractors for polynomial images of varieties, that is, distributions sampled by applying a low-degree polynomial map $f : \mathbb{F}_q^r \to \mathbb{F}_q^n$ to an element sampled uniformly at random from a $k$-dimensional variety $V \subseteq \mathbb{F}_q^r$. This class of sources generalizes both polynomial sources, studied by Dvir, Gabizon and Wigderson (FOCS 2007, Comput. Complex. 2009), and variety sources, studied by Dvir (CCC 2009, Comput. Complex. 2012). Assuming certain natural non-degeneracy conditions on the map $f$ and the variety $V$, which in particular ensure that the source has enough min-entropy, we extract almost all the min-entropy of the distribution. Unlike the Dvir-Gabizon-Wigderson and Dvir results, our construction works over large enough finite fields of arbitrary characteristic. One key part of our construction is an improved deterministic rank extractor for varieties. As a by-product, we obtain explicit Noether normalization lemmas for affine varieties and affine algebras. Additionally, we generalize a construction of affine extractors with exponentially small error due to Bourgain, Dvir and Leeman (Comput. Complex. 2016) by extending it to all finite prime fields of quasipolynomial size.