Derandomization from Algebraic Hardness

A hitting-set generator (HSG) is a polynomial map G:𝔽^k →𝔽^n such that for all n-variate polynomials C of small enough circuit size and degree, if C is nonzero, then C∘ G is nonzero. In this paper, we give a new construction of such an HSG assuming that we have an explicit polynomial of sufficient hardness. Formally, we prove the following over any field of characteristic zero: Let k∈ℕ and δ > 0 be arbitrary constants. Suppose {P_d}_d∈ℕ is an explicit family of k-variate polynomials such that deg P_d = d and P_d requires algebraic circuits of size d^δ. Then, there are explicit hitting sets of polynomial size for 𝖵𝖯. This is the first HSG in the algebraic setting that yields a complete derandomization of polynomial identity testing (PIT) for general circuits from a suitable algebraic hardness assumption. As a direct consequence, we show that even saving a single point from the "trivial" explicit, exponential sized hitting sets for constant-variate polynomials of low individual degree which are computable by small circuits, implies a deterministic polynomial time algorithm for PIT. More precisely, we show the following: Let k∈ℕ and δ > 0 be arbitrary constants. Suppose for every s large enough, there is an explicit hitting set of size at most ((s+1)^k - 1) for the class of k-variate polynomials of individual degree s that are computable by size s^δ circuits. Then there is an explicit hitting set of size poly(s) for the class of s-variate polynomials, of degree s, that are computable by size s circuits. As a consequence, we give a deterministic polynomial time construction of hitting sets for algebraic circuits, if a strengthening of the τ-Conjecture of Shub and Smale is true.