Black-box Identity Testing for Low Degree Unmixed $ΣΠΣΠ(k)$ Circuits

A ΣΠΣΠ(k) circuit C = ∑k i=1 Fi = ∑k i=1 ∏di j=1 fij is unmixed if for each i ∈ [k], Fi = fi1(x1) · · · fin(xn), where each fij 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 ΣΠΣΠ(k) circuits. In order to obtain the black-box algorithm, we first show that a special class of low degree unmixed ΣΠΣΠ(k) circuits of size s is sO(k 2)-sparse. Then we construct a hitting set H in polynomial time for the low degree unmixed ΣΠΣΠ(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 ∈ H. This is the first polynomial time black-box algorithm for the low degree unmixed ΣΠΣΠ(k) circuits, which also partly answers a question of Saxena [16].