Polyhedral faces in Gram spectrahedra of binary forms

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2021
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Zusammenfassung

The positive semidefinite Gram matrices of a form f with real coefficients parametrize the sum-of-squares representations of f. The convex body formed by the entirety of these matrices is the so-called Gram spectrahedron of f. We analyze the facial structures of symmetric and Hermitian Gram spectrahedra in the case of binary forms. We give upper bounds for the dimensions of polyhedral faces in Hermitian Gram spectrahedra and show that, if the form f is sufficiently generic, they can be realized by faces that are simplices and whose extreme points are rank-one tensors. We use our construction to prove a similar statement for the real symmetric case.

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Fachgebiet (DDC)
510 Mathematik
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Convex algebraic geometry; Sums of squares; Spectrahedra; Face; Binary form; Polyhedra
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ISO 690MAYER, Thorsten, 2021. Polyhedral faces in Gram spectrahedra of binary forms. In: Linear Algebra and Its Applications. Elsevier. 2021, 608, pp. 133-157. ISSN 0024-3795. eISSN 1873-1856. Available under: doi: 10.1016/j.laa.2020.08.025
BibTex
@article{Mayer2021Polyh-51726,
  year={2021},
  doi={10.1016/j.laa.2020.08.025},
  title={Polyhedral faces in Gram spectrahedra of binary forms},
  volume={608},
  issn={0024-3795},
  journal={Linear Algebra and Its Applications},
  pages={133--157},
  author={Mayer, Thorsten}
}
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