An Improved Construction of Variety-Evasive Subspace Families
This work provides a better explicit construction for a pseudorandom primitive that generalizes hitting sets and lossless rank condensers, relevant to theoretical computer science and algebraic complexity.
The authors construct explicit variety-evasive subspace families that evade all degree-d varieties in n-dimensional space, improving on previous constructions and coming within a polynomial factor of the lower bound for varieties of degree n^{1+Ω(1)}.
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.