Variety Evasive Subspace Families
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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 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.
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