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.

翻译：我们研究显式构造簇规避子空间族的问题，这是由Guo（计算复杂性，2024年）引入的一种伪随机原语，它同时推广了命中集和无损秩压缩器。粗略地说，簇规避子空间族$\mathcal{H}$是一个子空间的集合，使得对于固定族$\mathcal{F}$中的每个代数簇$V$，存在某个子空间$W \in \mathcal{H}$与$V$处于一般位置。我们给出了在$n$维仿射或射影空间中规避所有$d$次簇的子空间族的显式构造。我们的构造改进了Guo所构造的簇规避子空间族的大小，对于次数为$n^{1 + Ω(1)}$的簇，其大小与Guo关于任何此类簇规避子空间族大小的下界仅相差一个多项式因子。我们的簇规避子空间族依赖于对代数簇Chow形式的命中集的改进构造。