Uniform, Integral, and Feasible Proofs for the Determinant Identities

Abstract

Aiming to provide weak as possible axiomatic assumptions in which one can develop basic linear algebra, we give a uniform and integral version of the short propositional proofs for the determinant identities demonstrated over GF(2) in Hrubeš-Tzameret [15]. Specifically, we show that the multiplicativity of the determinant function and the Cayley-Hamilton theorem over the integers are provable in the bounded arithmetic theory VNC²; the latter is a first-order theory corresponding to the complexity class NC² consisting of problems solvable by uniform families of polynomial-size circuits and O(log² n)-depth. This also establishes the existence of uniform polynomial-size propositional proofs operating with NC²-circuits of the basic determinant identities over the integers (previous propositional proofs hold only over the two-element field).