Randomized polynomial time identity testing for noncommutative circuits

Abstract

In this paper we show that black-box polynomial identity testing for noncommutative polynomials f∈𝔽⟨z1,z2,…,zn⟩ of degree D and sparsity t, can be done in randomized (n,logt,logD) time. As a consequence, given a circuit C of size s computing a polynomial f∈𝔽⟨ z1,z2,…,zn⟩ with at most t non-zero monomials, then testing if f is identically zero can be done by a randomized algorithm with running time polynomial in s and n and logt. This makes significant progress on a question that has been open for over ten years. Our algorithm is based on automata-theoretic ideas that can efficiently isolate a monomial in the given polynomial. In particular, we carry out the monomial isolation using nondeterministic automata. In general, noncommutative circuits of size s can compute polynomials of degree exponential in s and number of monomials double-exponential in s. In this paper, we consider a natural class of homogeneous noncommutative circuits, that we call +-regular circuits, and give a white-box polynomial time deterministic polynomial identity test. These circuits can compute noncommutative polynomials with number of monomials double-exponential in the circuit size. Our algorithm combines some new structural results for +-regular circuits with known results for noncommutative ABP identity testing, rank bound of commutative depth three identities, and equivalence testing problem for words. Finally, we consider the black-box identity testing problem for depth three +-regular circuits and give a randomized polynomial time identity test. In particular, we show if f∈𝔽Z⟩ is a nonzero noncommutative polynomial computed by a depth three +-regular circuit of size s, then f cannot be a polynomial identity for the matrix algebra 𝕄s(𝔽) when 𝔽 is sufficiently large depending on the degree of f.