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Variety Evasive Subspace Families

Author Zeyu Guo

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Zeyu Guo
  • Department of Computer Science, University of Haifa, Israel

Acknowledgements

We thank Nitin Saxena, Noga Ron-Zewi, Amit Sinhababu, and Suryajith Chillara for helpful discussions. We also thank the anonymous reviewers of CCC'21 for their careful reading of our manuscript and their insightful suggestions that helped improve this paper.

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Zeyu Guo. Variety Evasive Subspace Families. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 20:1-20:33, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.CCC.2021.20

Abstract

We introduce the problem of constructing explicit variety evasive subspace families. Given a family ℱ of subvarieties of a projective or affine space, a collection ℋ of projective or affine k-subspaces is (ℱ,ε)-evasive if for every 𝒱 ∈ ℱ, all but at most ε-fraction of W ∈ ℋ intersect every irreducible component of 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 bounded 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 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • algebraic complexity
  • dimension reduction
  • Noether normalization
  • polynomial identity testing
  • pseudorandomness
  • varieties

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