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 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 (log 2 n )-depth. This also establishes the existence of uniform polynomial-size propositional proofs operating with NC 2 -circuits of the basic determinant identities over the integers (previous propositional proofs hold only over the two-element field).

中文翻译：

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行列式恒等式的统一、完整和可行证明

为了提供一个可以开发基本线性代数的尽可能弱的公理假设，我们给出了一个统一的和完整的版本的短命题证明，用于证明的行列式恒等式GF(2) 在 Hrubeš-Tzameret [15]。具体来说，我们证明了行列式函数的乘法性和整数上的 Cayley-Hamilton 定理在有界算术理论中是可证明的VNC 2; 后者是对应于复杂度类的一阶理论数控 2由多项式大小电路的统一族可解决的问题和○（日志2 n）-深度。这也建立了统一多项式大小的命题证明的存在数控 2- 整数上的基本行列式恒等式电路（以前的命题证明仅适用于二元域）。