Variety Evasive Sets

TL;DR

This paper constructs large subsets of finite field vector spaces that minimally intersect with any fixed-dimension, bounded-degree affine variety, extending previous work on affine subspaces.

Contribution

It provides an explicit construction of evasive sets in finite fields that generalize prior results limited to affine subspaces.

Findings

Constructs large subsets with small intersection properties

Generalizes previous results from affine subspaces to higher-degree varieties

Offers explicit methods for building such evasive sets

Abstract

We give an explicit construction of a large subset of F^n, where F is a finite field, that has small intersection with any affine variety of fixed dimension and bounded degree. Our construction generalizes a recent result of Dvir and Lovett (STOC 2012) who considered varieties of degree one (affine subspaces).