paper

On the dimension of the locus of determinantal hypersurfaces

arXiv:1602.08623 · doi:10.4153/CMB-2016-070-8

Abstract

The characteristic polynomial of an $r$ -tuple $(A_{1}, ..., A_{r})$ of $n \times n$ matrices is the determinant $det (x_{0} I + x_{1} A_{1} + ... + x_{r} A_{r})$ . We show that if $r$ is at least 3 and $A = (A_{1}, ..., A_{r})$ is an $r$ -tuple of matrices in general position, then up to conjugacy there are only finitely many $r$ -tuples of matrices with the same characteristic polynomial as $A$ . Equivalently, the locus of determinantal hypersurfaces of degree $n$ in $P^{r}$ is irreducible of dimension $(r - 1) n^{2} + 1$ .

18 pages. Added a section