A Case of Depth-3 Identity Testing, Sparse Factorization and Duality

Abstract

The polynomial identity testing (PIT) problem is known to be challenging even for constant depth arithmetic circuits. In this work, we study the complexity of two special but natural cases of identity testing--first is a case of depth-3 PIT, the other of depth-4 PIT. Our first problem is a vast generalization of verifying whether a bounded top fan-in depth-3 circuit equals a sparse polynomial (given as a sum of monomial terms). Formally, given a depth-3 circuit C, having constant many general product gates and arbitrarily many semidiagonal product gates test whether the output of C is identically zero. A semidiagonal product gate in C computes a product of the form $${m \cdot \prod^b_{i=1}l^{e_i}_i}$$ , where m is a monomial, l i is a linear polynomial, and b is a constant. We give a deterministic polynomial time test, along with the computation of leading monomials of semidiagonal circuits over local rings. The second problem is on verifying a given sparse polynomial factorization, which is a classical question (von zur Gathen, FOCS 1983): Given multivariate sparse polynomials f, g1, . . . , gt explicitly check whether $${f = \prod^t_{i=1} {g_i}}$$ . For the special case when every gi is a sum of univariate polynomials, we give a deterministic polynomial time test. We characterize the factors of such g i 's and even show how to test the divisibility of f by the powers of such polynomials. The common tools used are Chinese remaindering and dual representation. The dual representation of polynomials (Saxena, ICALP 2008) is a technique to express a product-of-sums of univariates as a sum-ofproducts of univariates. We generalize this technique by combining it with a generalized Chinese remaindering to solve these two problems (over any field).