Generating Matrix Identities and Hard Instances for Strong Proof Systems

We study the complexity of generating identities of matrix rings. We establish an unconditional lower bound on the minimal number of generators needed to generate a matrix identity, where the generators are substitution instances of elements from any given finite basis of the matrix identities. Based on our findings, we propose to consider matrix identities (and their encoding) as hard instances for strong proof systems, and we initiate this study under different settings and assumptions. We show that this direction, under certain conditions, can potentially lead up to exponential-size lower bounds on arithmetic proofs, which are proofs of polynomial identities operating with arithmetic circuits and whose axioms are the polynomial-ring axioms (these proofs serve as an algebraic analogue of the Extended Frege propositional proof system). We also discuss shortly the applicability of our approach to strong propositional proof systems. Formally, the algebraic problem we study is this: for a field F let A be a non-commutative (associative) F-algebra (e.g., the algebra Matd(F) of d × d matrices over F). We say that a non-commutative polynomial f(x1, . . . , xn) over F is an identity of A, if for all c ∈ A, f(c) = 0. Let B be a set of non-commutative polynomials that forms a basis for the identities of A, in the following sense: for every identity f of A there exist non-commutative polynomials g1, . . . , gk, for some k, that are substitution instances of polynomials from B, such that f is in the (two-sided) ideal 〈g1, . . . , gk〉. We ask the following question: Given A,B and f as above, what is the minimal number k of such generators g1, . . . , gk for which f ∈ 〈g1, . . . , gk〉? In particular, we focus on the case where the algebra A is Matd(F), and F has characteristic 0. Our main technical contribution is a generalization of the lower bound presented in Hrubeš [7] (for the case d = 1) to any d > 2: • For every natural number d > 2 and every finite basis B for the identities of Matd(F), where F is of characteristic 0, there exists an identity fn with n variables, that requires Ω(n) generators (i.e., substitution instances from B) to generate. The proof uses fundamental results from the theory of algebras with polynomial identities (PI-algebras) together with a generalization of the arguments in [7].