Variety Evasive Subspace Families

TL;DR

This paper introduces the concept of variety evasive subspace families, constructs explicit examples using Chow forms, and applies these to derandomize Noether's normalization and improve polynomial identity testing algorithms.

Contribution

It provides the first explicit constructions of variety evasive subspace families and demonstrates their applications in derandomization and PIT for specific circuit classes.

Findings

Explicit construction of variety evasive families using Chow forms.

Complete derandomization of Noether's normalization for low-degree varieties.

Polynomial-time black-box PIT algorithm for certain depth-4 circuits.

Abstract

We introduce the problem of constructing explicit variety evasive subspace families. Given a family $\mathcal{F}$ of subvarieties of a projective or affine space, a collection $\mathcal{H}$ of projective or affine $k$-subspaces is $(\mathcal{F},\epsilon)$-evasive if for every $\mathcal{V}\in\mathcal{F}$, all but at most $\epsilon$-fraction of $W\in\mathcal{H}$ intersect every irreducible component of $\mathcal{V}$ with (at most) the expected dimension. The problem of constructing such an explicit subspace family generalizes both deterministic black-box polynomial identity testing (PIT) and the problem of constructing explicit (weak) lossless rank condensers. Using Chow forms, we construct explicit $k$-subspace families of polynomial size that are evasive for all varieties of bounded degree in a projective or affine $n$-space. As one application, we obtain a complete derandomization of Noether's normalization lemma for varieties of low degree in a projective or affine $n$-space. In another application, we obtain a simple polynomial-time black-box PIT algorithm for depth-4 arithmetic circuits with bounded top fan-in and bottom fan-in that are not in the Sylvester-Gallai configuration, improving and simplifying a result of Gupta (ECCC TR 14-130). As a complement of our explicit construction, we prove a tight lower bound for the size of $k$-subspace families that are evasive for degree-$d$ varieties in a projective $n$-space. When $n-k=n^{\Omega(1)}$, the lower bound is superpolynomial unless $d$ is bounded. The proof uses a dimension-counting argument on Chow varieties that parametrize projective subvarieties.