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URN: urn:nbn:de:0030-drops-142949
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Variety Evasive Subspace Families

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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.

BibTeX - Entry

@InProceedings{guo:LIPIcs.CCC.2021.20,
  author =	{Guo, Zeyu},
  title =	{{Variety Evasive Subspace Families}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{20:1--20:33},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-193-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{200},
  editor =	{Kabanets, Valentine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/14294},
  URN =		{urn:nbn:de:0030-drops-142949},
  doi =		{10.4230/LIPIcs.CCC.2021.20},
  annote =	{Keywords: algebraic complexity, dimension reduction, Noether normalization, polynomial identity testing, pseudorandomness, varieties}
}

Keywords: algebraic complexity, dimension reduction, Noether normalization, polynomial identity testing, pseudorandomness, varieties
Seminar: 36th Computational Complexity Conference (CCC 2021)
Issue date: 2021
Date of publication: 08.07.2021


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