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Variety Evasive Subspace Families

Zeyu Guo

TL;DR

This work introduces variety evasive subspace families as a nonlinear generalization of hitting sets for deterministic PIT, focusing on subvarieties of bounded degree in projective/affine space. The authors develop an explicit, polynomial-size construction of $(n,d,\varepsilon)$-evasive $k$-subspace families using Chow forms and a two-step projection approach, enabling derandomization of Noether's normalization lemma for low-degree varieties and a simple polynomial-time PIT for depth-4 circuits outside Sylvester-Gallai configurations. A tight lower bound via Chow varieties shows inherent size limitations when $n-k$ is large and $d$ grows, highlighting a gap between upper and lower bounds in general. The results advance nonlinear pseudorandomness, connect algebraic geometry with derandomization, and yield practical PIT improvements for structured circuit classes with potential broader extractor implications.

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 $(\mathcal{F},ε)$-evasive if for every $\mathcal{V}\in\mathcal{F}$, all but at most $ε$-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 $n-k=n^{Ω(1)}$, the lower bound is superpolynomial unless $d$ is bounded. The proof uses a dimension-counting argument on Chow varieties that parametrize projective subvarieties.

Variety Evasive Subspace Families

TL;DR

This work introduces variety evasive subspace families as a nonlinear generalization of hitting sets for deterministic PIT, focusing on subvarieties of bounded degree in projective/affine space. The authors develop an explicit, polynomial-size construction of -evasive -subspace families using Chow forms and a two-step projection approach, enabling derandomization of Noether's normalization lemma for low-degree varieties and a simple polynomial-time PIT for depth-4 circuits outside Sylvester-Gallai configurations. A tight lower bound via Chow varieties shows inherent size limitations when is large and grows, highlighting a gap between upper and lower bounds in general. The results advance nonlinear pseudorandomness, connect algebraic geometry with derandomization, and yield practical PIT improvements for structured circuit classes with potential broader extractor implications.

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 Noether's normalization lemma for varieties of low degree in a projective or affine -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 -subspace families that are evasive for degree- varieties in a projective -space. When , the lower bound is superpolynomial unless is bounded. The proof uses a dimension-counting argument on Chow varieties that parametrize projective subvarieties.

Paper Structure

This paper contains 56 sections, 31 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Variety Evasive Subspace Families
  3. Connection with hitting sets.
  4. Connection with lossless rank condensers.
  5. Our Results
  6. Lower bound.
  7. Derandomizing Noether's Normalization Lemma
  8. Dimension-preserving morphisms vs. finite morphisms.
  9. Depth-4 Polynomial Identity Testing
  10. Proof Overview
  11. Overview of the proof of \ref{['thm_main']}.
  12. Reducing to the equidimensional/irreducible case of dimension $n-k-1$.
  13. Hitting the Chow form of $\mathcal{V}$.
  14. A two-step construction.
  15. Overview of the proof of \ref{['thm_lb']}.
  16. ...and 41 more sections

Key Result

Theorem 1.6

For $n,d\in\mathbb{N}^+$, $k\in\{0,1,\dots,n-1\}$, and $\varepsilon\in (0,1)$, there exists an $(n, d, \varepsilon)$-evasive $k$-subspace family (resp. affine $k$-subspace family) $\mathcal{H}$ on $\mathbb{P}^n$ (resp. $\mathbb{A}^n$) of size $\mathrm{poly}(N(k,d,n), n, 1/\varepsilon)$, which is bou In particular, when $d$ is bounded, the bit complexity of constructing $\mathcal{H}$ is polynomial

Theorems & Definitions (69)

  • Definition 1.1: Evading
  • Definition 1.2: Subspace family
  • Definition 1.3: Variety evasive subspace family
  • Definition 1.4: degree
  • Definition 1.5
  • Remark
  • Theorem 1.6: Main Theorem
  • Remark
  • Remark
  • Theorem 1.7
  • ...and 59 more