## Annals of Functional Analysis

### Computation of Riemann matrices for the hyperbolic curves of determinantal polynomials

#### Abstract

The numerical range of a matrix, according to Kippenhahn, is determined by a hyperbolic determinantal form of linear Hermitian matrices associated to the matrix. On the other hand, using Riemann theta functions, Helton and Vinnikov confirmed that a hyperbolic form always admits a determinantal representation of linear real symmetric matrices. The Riemann matrix of the hyperbolic curve plays the main role in the existence of real symmetric matrices. In this article, we implement computations of the Riemann matrix and the Abel–Jacobi variety of the hyperbolic curve associated to a determinantal polynomial of a matrix. Further, we prove that the lattice of the Abel–Jacobi variety is decomposed into the direct sum of two orthogonal lattices for some $4\times4$ Toeplitz matrices.

#### Article information

Source
Ann. Funct. Anal., Volume 8, Number 2 (2017), 152-167.

Dates
Accepted: 12 August 2016
First available in Project Euclid: 14 January 2017

https://projecteuclid.org/euclid.afa/1484363067

Digital Object Identifier
doi:10.1215/20088752-3773229

Mathematical Reviews number (MathSciNet)
MR3597154

Zentralblatt MATH identifier
1360.15024

#### Citation

Chien, Mao-Ting; Nakazato, Hiroshi. Computation of Riemann matrices for the hyperbolic curves of determinantal polynomials. Ann. Funct. Anal. 8 (2017), no. 2, 152--167. doi:10.1215/20088752-3773229. https://projecteuclid.org/euclid.afa/1484363067

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