Uniform, Integral and Feasible Proofs for the Determinant Identities
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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 VNC^2; the latter is a first-order theory corresponding to the complexity class NC^2 consisting of problems solvable by uniform families of polynomial-size circuits and O( ^2 n)-depth. This also establishes the existence of uniform polynomial-size NC^2-Frege proofs of the basic determinant identities over the integers (previous propositional proofs hold only over the two element field).
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