Uniform, Integral and Feasible Proofs for the Determinant Identities

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 Hrubes-Tzameret [SICOMP'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 $\mathbf{VNC}^2$; the latter is a first-order theory corresponding to the complexity class $\mathbf{NC}^2$ consisting of problems solvable by uniform families of polynomial-size circuits and $O(\log ^2 n)$-depth. This also establishes the existence of uniform polynomial-size $\mathbf{NC}^2$-Frege proofs of the basic determinant identities over the integers (previous propositional proofs hold only over the two element field).