Journal of Convex Analysis 33 (2026), No. 1&2, 227--242
Copyright Heldermann Verlag 2026
Minimal Hyperbolic Polynomials and Ranks of Homogeneous Cones
Joao Gouveia
CMUC, Dept. of Mathematics, University of Coimbra, Coimbra, Portugal
jgouveia@mat.uc.pt
Masaru Ito
Dept. of Mathematics, College of Science and Technology, Nihon University, Tokyo, Japan
ito.masaru@nihon-u.ac.jp
Bruno F. Louren
Dept. of Fundamental Statistical Mathematics, Institute of Statistical Mathematics, Kyoto University, Kyoto, Japan
bruno@ism.ac.jp
The starting point of this paper is the computation of minimal hyperbolic polynomials of duals of cones arising from chordal sparsity patterns. From that, we investigate the relation between ranks of homogeneous cones and their minimal polynomials. Along the way, we answer in the negative a question posed in an earlier paper and show examples of homogeneous cones that cannot be realized as rank-one generated (ROG) hyperbolicity cones.
Keywords: Hyperbolic polynomial, homogeneous cone, hyperbolicity cone, chordal graphs, minimal polynomial.
MSC: 52A20, 14P10, 05C50.
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