The polynomial identity lemma (also called the "Schwartz-Zippel lemma") states that any nonzero polynomial $f(x_1,\ldots, x_n)$ of degree at most $s$ will evaluate to a nonzero value at some point on a grid $S^n \subseteq \mathbb{F}^n$ with $|S| > s$. Thus, there is an explicit hitting set for all $n$-variate degree $s$, size $s$ algebraic circuits of size $(s+1)^n$. In this paper, we prove the following results: - Let $\varepsilon > 0$ be a constant. For a sufficiently large constant $n$ and all $s > n$, if we have an explicit hitting set of size $(s+1)^{n-\varepsilon}$ for the class of $n$-variate degree $s$ polynomials that are computable by algebraic circuits of size $s$, then for all $s$, we have an explicit hitting set of size $s^{\exp \circ \exp (O(\log^\ast s))}$ for $s$-variate circuits of degree $s$ and size $s$. That is, if we can obtain a barely non-trivial exponent compared to the trivial $(s+1)^{n}$ sized hitting set even for constant variate circuits, we can get an almost complete derandomization of PIT. - The above result holds when "circuits" are replaced by "formulas" or "algebraic branching programs". This extends a recent surprising result of Agrawal, Ghosh and Saxena (STOC 2018,PNAS 2019) who proved the same conclusion for the class of algebraic circuits, if the hypothesis provided a hitting set of size at most $(s^{n^{0.5 - \delta}})$ (where $\delta>0$ is any constant). Hence, our work significantly weakens the hypothesis of Agrawal, Ghosh and Saxena to only require a slightly non-trivial saving over the trivial hitting set, and also presents the first such result for algebraic branching programs and formulas.

翻译：多项式恒等式引理（亦称"施瓦茨-齐普尔引理"）指出：任意次数至多为$s$的非零多项式$f(x_1,\ldots, x_n)$在网格$S^n \subseteq \mathbb{F}^n$（其中$|S| > s$）上某点的求值结果必非零。因此，对于全体$n$变量次数$s$、规模$s$的代数电路，存在一个显式命中集，其大小为$(s+1)^n$。本文证明如下结果： - 令$\varepsilon > 0$为常数。对于充分大的常数$n$及所有$s > n$，若我们对可被规模$s$的代数电路计算的$n$变量次数$s$多项式类，存在一个大小为$(s+1)^{n-\varepsilon}$的显式命中集，则对所有$s$，我们都能得到大小为$s^{\exp \circ \exp (O(\log^\ast s))}$的显式命中集，用于处理$s$变量次数$s$且规模$s$的电路。这意味着，即使对于常变量电路，只要能相比平凡命中集$(s+1)^{n}$实现微弱的非平凡指数改进，就可以近乎完全地实现PIT的随机性消解。 - 上述结论在将"电路"替换为"公式"或"代数分支程序"时依然成立。这扩展了Agrawal、Ghosh与Saxena（STOC 2018, PNAS 2019）近期令人惊讶的结果——他们针对代数电路类证明了相同结论，但要求假设条件提供的命中集大小至多为$(s^{n^{0.5 - \delta}})$（其中$\delta>0$为任意常数）。因此，本文显著弱化了Agrawal、Ghosh与Saxena的假设条件，仅需相对平凡命中集实现略微非平凡的节省，同时首次给出代数分支程序与公式类的此类结果。