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arXiv · 1711.01106

On determinantal ideals and algebraic dependence

Margherita Barile·Antonio Macchia
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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.

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Margherita Barile, Antonio Macchia. 2017-11-03. On determinantal ideals and algebraic dependence. https://arxiv.org/abs/1711.01106

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