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### Paper:

TR18-132 | 17th July 2018 12:56

#### Near-optimal Bootstrapping of Hitting Sets for Algebraic Circuits

TR18-132
Authors: Mrinal Kumar, Ramprasad Saptharishi, Anamay Tengse
Publication: 22nd July 2018 16:05
The classical lemma of Ore-DeMillo-Lipton-Schwartz-Zippel 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 a deterministic polynomial identity test (PIT) for all degree-$s$ size-$s$ algebraic circuits in $n$ variables that runs in time $\mathrm{poly}(s) \cdot (s+1)^n$.
In a surprising recent result, Agrawal, Ghosh and Saxena (STOC 2018) showed any deterministic blackbox PIT algorithm for degree-$s$, size-$s$, $n$-variate circuits with running time as bad as $s^{n^{0.5 - \delta}}\mathrm{Huge}(n)$, where $\delta > 0$ and $\mathrm{Huge}(n)$ is an arbitrary function, can be used to construct blackbox PIT algorithms for degree-$s$ size $s$ circuits with running time $s^{\exp \circ \exp (O(\log ^\ast s))}$.
The authors asked if a similar conclusion followed if their hypothesis was weakened to having deterministic PIT with running time $s^{o(n)}\cdot \mathrm{Huge}(n)$. In this paper, we answer their question in the affirmative. We show that, given a deterministic blackbox PIT that runs in time $s^{o(n)}\cdot \mathrm{Huge}(n)$ for all degree-$s$ size-$s$ algebraic circuits over $n$ variables, we can obtain a deterministic blackbox PIT that runs in time $s^{\exp \circ \exp(O(\log^{*}s))}$ for all degree-$s$ size-$s$ algebraic circuits over $n$ variables. In other words, any blackbox PIT with just a slightly non-trivial exponent of $s$ compared to the trivial $s^{O(n)}$ test can be used to give a nearly polynomial time blackbox PIT algorithm.