An Improved Construction of Variety-Evasive Subspace Families

TL;DR

This paper presents an explicit construction of variety-evasive subspace families that improve size bounds and approach theoretical lower bounds, advancing pseudorandom primitive design.

Contribution

It introduces a new construction method for variety-evasive subspace families that outperforms previous work and nearly matches lower bounds for large-degree varieties.

Findings

Constructed subspace families that evade all degree-d varieties.

Improved size bounds over previous constructions.

Approaches Guo's lower bound for large-degree varieties.

Abstract

We study the question of explicitly constructing variety-evasive subspace families, a pseudorandom primitive introduced by Guo (Computational Complexity 2024) that generalizes both hitting sets and lossless rank condensers. Roughly speaking, a variety-evasive subspace family $\mathcal{H}$ is a collection of subspaces such that for every algebraic variety $V$ in a fixed family $\mathcal{F}$, there is some subspace $W \in \mathcal{H}$ that is in general position with respect to $V$. We give an explicit construction of a subspace families that evade all degree-$d$ varieties in an $n$-dimensional affine or projective space. Our construction improves on the size of the variety-evasive subspace families constructed by Guo and, for varieties of degree $n^{1 + \Omega(1)}$, comes within a polynomial factor of Guo's lower bound on the size of any such variety-evasive subspace family. Our variety-evasive subspace families rely on an improved construction of hitting sets for Chow forms of algebraic varieties.