Derandomization from Algebraic Hardness

A hitting-set generator (HSG) is a polynomial map $G:\mathbb{F}^k \to \mathbb{F}^n$ such that for all $n$-variate polynomials $C$ of small enough circuit size and degree, if $C$ is nonzero, then $C\circ G$ 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 $k\in \mathbb{N}$ and $\delta > 0$ be arbitrary constants. Suppose $\{P_d\}_{d\in \mathbb{N}}$ is an explicit family of $k$-variate polynomials such that $\operatorname{deg} P_d = d$ and $P_d$ requires algebraic circuits of size $d^\delta$. Then, there are explicit hitting sets of polynomial size for $\mathsf{VP}$. 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 $k\in \mathbb{N}$ and $\delta > 0$ 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 $s^\delta$ circuits. Then there is an explicit hitting set of size $\operatorname{poly}(s)$ 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 $\tau$-Conjecture of Shub and Smale is true.