Variety Evasive Subspace Families

Abstract

AbstractWe introduce the problem of constructing explicit variety evasive subspace families. Given a family $$\mathcal{F}$$ F of subvarieties of a projective or affine space, a collection $$\mathcal{H}$$ H of projective or affine $$k$$ k -subspaces is $$(\mathcal{F},\epsilon)$$ ( F , ϵ ) -evasive if for every $$\mathcal{V}\in\mathcal{F}$$ V ∈ F , all but at most $$\epsilon$$ ϵ -fraction of $$W\in\mathcal{H}$$ W ∈ H intersect every irreducible component of $$\mathcal{V}$$ 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 $$k$$ k -subspace families of polynomial size that are evasive for all varieties of bounded degree in a projective or affine $$n$$ 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 $$n$$ 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 $$k$$ k -subspace families that are evasive for degree-$$d$$ d varieties in a projective $$n$$ n -space. When $$n-k=n^{\Omega(1)}$$ n - k = n Ω ( 1 ) , the lower bound is superpolynomial unless $$d$$ d is bounded. The proof uses a dimension counting argument on Chow varieties that parametrize projective subvarieties.