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# On the Dimension of the Locus of Determinantal Hypersurfaces

Published:2016-11-11
Printed: Sep 2017
• Zinovy Reichstein,
Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada V6T 1Z2
• Angelo Vistoli,
Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy
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## Abstract

The characteristic polynomial $P_A(x_0, \dots, x_r)$ of an $r$-tuple $A := (A_1, \dots, A_r)$ of $n \times n$-matrices is defined as $P_A(x_0, \dots, x_r) := \det(x_0 I + x_1 A_1 + \dots + x_r A_r) \, .$ We show that if $r \geqslant 3$ and $A := (A_1, \dots, A_r)$ is an $r$-tuple of $n \times n$-matrices in general position, then up to conjugacy, there are only finitely many $r$-tuples $A' := (A_1', \dots, A_r')$ such that $p_A = p_{A'}$. Equivalently, the locus of determinantal hypersurfaces of degree $n$ in $\mathbf{P}^r$ is irreducible of dimension $(r-1)n^2 + 1$.
 Keywords: determinantal hypersurface, matrix invariant, $q$-binomial coefficient