On determinantal ideals and algebraic dependence

TL;DR

This paper investigates how algebraic dependence among matrix entries affects the arithmetical rank of determinantal ideals, establishing bounds that depend on the pattern of dependencies and zero entries, with sharp results in characteristic zero.

Contribution

It provides new bounds on the arithmetical rank of determinantal ideals based on algebraic dependencies and zero entries, extending understanding of their algebraic complexity.

Findings

Arithmetical rank drops by at least one when entries outside a t-minor are dependent.

Drop in arithmetical rank is at least k if the matrix has k zero entries under certain conditions.

Bounds are sharp in characteristic zero, matching local cohomological dimension.

Abstract

Let $X$ be a matrix with entries in a polynomial ring over an algebraically closed field $K$. We prove that, if the entries of $X$ outside some $(t \times t)$-submatrix are algebraically dependent over $K$, the arithmetical rank of the ideal $I_t(X)$ of $t$-minors of $X$ drops at least by one with respect to the generic case; under suitable assumptions, it drops at least by $k$ if $X$ has $k$ zero entries. This upper bound turns out to be sharp if $\mathrm{char}\, K=0$, since it then coincides with the lower bound provided by the local cohomological dimension.