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

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}.