Black-box Identity Testing for Low Degree Unmixed $\Sigma\Pi\Sigma\Pi(k)$ Circuits

TL;DR

This paper presents the first polynomial-time black-box identity testing algorithm for low degree unmixed (k) circuits, leveraging sparsity and constructing a polynomial-size hitting set.

Contribution

It introduces a novel sparsity bound for low degree unmixed (k) circuits and constructs an efficient hitting set for identity testing.

Findings

Polynomial-time black-box identity testing algorithm developed.

Sparsity bound of s^{O(k^2)} for certain circuits established.

Constructed polynomial-size hitting set for low degree unmixed (k) circuits.

Abstract

A $\Sigma\Pi\Sigma\Pi(k)$ circuit $C=\sum_{i=1}^kF_i=\sum_{i=1}^k\prod_{j=1}^{d_i}f_{ij}$ is unmixed if for each $i\in[k]$, $F_i=f_{i1}(x_1)... f_{in}(x_n)$, where each $f_{ij}$ is a univariate polynomial given in the sparse representation. In this paper, we give a polynomial time black-box algorithm of identity testing for the low degree unmixed $\Sigma\Pi\Sigma\Pi(k)$ circuits. In order to obtain the black-box algorithm, we first show that a special class of low degree unmixed $\Sigma\Pi\Sigma\Pi(k)$ circuits of size $s$ is $s^{O(k^2)}$-sparse. Then we construct a hitting set $\mathcal{H}$ in polynomial time for the low degree unmixed $\Sigma\Pi\Sigma\Pi(k)$ circuits from the sparsity result above. The constructed hitting set is polynomial size. Thus we can test whether the circuit or the polynomial $C$ is identically zero by checking whether $C(a)=0$ for each $a\in\mathcal{H}$. This is the first polynomial time black-box algorithm for the low degree unmixed $\Sigma\Pi\Sigma\Pi(k)$ circuits, which also partly answers a question of Saxena \cite{SAX}.