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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 determinantal hypersurface, matrix invariant, $q$-binomial coefficient
MSC Classifications: 14M12, 15A22, 05A10 show english descriptions Determinantal varieties [See also 13C40]
Matrix pencils [See also 47A56]
Factorials, binomial coefficients, combinatorial functions [See also 11B65, 33Cxx]
14M12 - Determinantal varieties [See also 13C40]
15A22 - Matrix pencils [See also 47A56]
05A10 - Factorials, binomial coefficients, combinatorial functions [See also 11B65, 33Cxx]
 

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