Deterministic polynomial identity tests for multilinear bounded-read formulae

Abstract

We present a polynomial-time deterministic algorithm for testing whether constant-read multilinear arithmetic formulae are identically zero. In such a formula, each variable occurs only a constant number of times, and each subformula computes a multilinear polynomial. Our algorithm runs in time $${s^{O(1)}\cdot n^{k^{O(k)}}}$$sO(1)·nkO(k) , where s denotes the size of the formula, n denotes the number of variables, and k bounds the number of occurrences of each variable. Before our work, no subexponential-time deterministic algorithm was known for this class of formulae. We also present a deterministic algorithm that works in a blackbox fashion and runs in time $${n^{k^{O(k)} + O(k \log n)}}$$nkO(k)+O(klogn) in general, and time $${n^{k^{O(k^2)} + O(kD)}}$$nkO(k2)+O(kD) for depth D. Finally, we extend our results and allow the inputs to be replaced with sparse polynomials. Our results encompass recent deterministic identity tests for sums of a constant number of read-once formulae and for multilinear depth-four formulae.