Variety Evasive Subspace Families

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