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).

为了提供一个可以发展基本线性代数的尽可能弱的公理假设，我们给出了在Hrubeš-Tzameret[15]中通过GF（2）证明的行列式身份的简短命题证明的统一和完整版本。具体来说，我们证明了行列式函数和Cayley-Hamilton定理在整数上的可乘性在有界算术理论VNC ^2中是可证明的。后者是一阶理论，对应于复杂度等级NC ^2，由可通过多项式大小电路的统一族和O（log ² n）-深度。这也建立了使用整数的基本行列式恒等式的NC ^2-回路的统一多项式大小的命题证明的存在（先前的命题证明仅适用于二元域）。