  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.
