An Improved Construction of Variety-Evasive Subspace Families

2026-05-06

AI Summary

This work addresses the explicit construction of variety-evasive subspace families—collections of subspaces that avoid all algebraic varieties from a given family—a problem of fundamental importance in pseudorandomness, hitting set constructions, and lossless rank condensers. By refining hitting set constructions based on Chow forms, we explicitly build subspace families in $n$-dimensional affine or projective space that evade all degree-$d$ algebraic varieties, with size nearly matching the theoretical lower bound. When $d = n^{1+\Omega(1)}$, our construction achieves a size exceeding the lower bound by only a polynomial factor, substantially improving upon prior work by Guo and yielding the best-known explicit construction to date.

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 $n^{1 + Ω(1)}$, 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.

Problem

• variety-evasive
• subspace families
• algebraic varieties
• explicit construction
• Chow forms

Innovation

• variety-evasive subspace families
• explicit construction
• hitting sets
• Chow forms
• algebraic varieties