Provability of determinant and matrix-rank properties in V#L

Determine whether the determinant properties—specifically the cofactor expansion, the axiomatic definition of the determinant (multilinearity, alternation, identity), and the Cayley–Hamilton Theorem—and the matrix-rank properties are provable within the bounded arithmetic theory V#L, rather than only in stronger theories such as VNC^2.

Background

The introduction reviews known results: several determinant properties (cofactor expansion, axiomatic definition, Cayley–Hamilton) are equivalent over V#L, and multiplicativity implies them. Tzameret and Cook [9] established that multiplicativity of the determinant is provable in VNC^2, and Ken and Kuroda [4] showed that matrix-rank properties are provable in VNC^2 via interpretations of Soltys’s linear algebra theory.

Despite these advances in the stronger theory VNC^2, the paper explicitly notes that it remains unresolved whether the aforementioned determinant properties and matrix-rank properties can be proved in the weaker bounded arithmetic theory V#L. This question motivates extending proof complexity studies from the determinant to the Pfaffian.