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 nonzero, then $C\circ G$ nonzero. In this paper, we give new construction an HSG assuming have explicit sufficient hardness. Formally, prove the following over any field characteristic zero: Let $k\in \mathbb{N}$ $\delta > 0$ be arbitrary constants. Suppose $\{P_d\}_{d\in \mathbb{N}}$ family $k$-variate $\operatorname{deg} P_d = d$ $P_d$ requires algebraic circuits $d^\delta$. Then, there are hitting sets $\mathsf{VP}$. This first in setting yields complete derandomization identity testing (PIT) general from suitable hardness assumption. As direct consequence, show even saving single point trivial explicit, exponential sized constant-variate low individual degree which computable by circuits, implies deterministic time algorithm PIT. More precisely, following: every $s$ large enough, set at most $((s+1)^k - 1)$ class $s^\delta$ circuits. Then $\operatorname{poly}(s)$ $s$-variate polynomials, $s$, As strengthening $\tau$-Conjecture Shub Smale true.