
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 of subvarieties of a projective or affine space, a collection of projective or affine -subspaces is -evasive if for every , all but at most -fraction of intersect every irreducible component of 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 -subspace families of polynomial size that are evasive for all varieties of bounded degree in a projective or affine -space. As one application, we obtain a complete derandomization of…
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