Computing the determinantal representations of hyperbolic forms
Czechoslovak Mathematical Journal, Tome 66 (2016) no. 3, pp. 633-651.

Voir la notice de l'article dans Czech Digital Mathematics Library

The numerical range of an $n\times n$ matrix is determined by an $n$ degree hyperbolic ternary form. Helton-Vinnikov confirmed conversely that an $n$ degree hyperbolic ternary form admits a symmetric determinantal representation. We determine the types of Riemann theta functions appearing in the Helton-Vinnikov formula for the real symmetric determinantal representation of hyperbolic forms for the genus $g=1$. We reformulate the Fiedler-Helton-Vinnikov formulae for the genus $g=0,1$, and present an elementary computation of the reformulation. Several examples are provided for computing the real symmetric matrices using the reformulation.
DOI : 10.1007/s10587-016-0283-9
Classification : 14Q05, 15A60
Mots-clés : determinantal representation; hyperbolic form; Riemann theta function; numerical range 
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Chien, Mao-Ting; Nakazato, Hiroshi. Computing the determinantal representations of hyperbolic forms. Czechoslovak Mathematical Journal, Tome 66 (2016) no. 3, pp. 633-651. doi : 10.1007/s10587-016-0283-9. https://geodesic-test.mathdoc.fr/articles/10.1007/s10587-016-0283-9/

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