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Hilbert’s 1888 Theorem for Cones along the Veronese Variety

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2024

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The cone of positive semidefinite homogeneous polynomials with real coefficients in n+1 variables of degree 2d contains the cone of those that are representable as finite sums of squares. Hilbert’s 1888 theorem states that these cones coincide exactly in the Hilbert cases of binary forms, quadratic forms or ternary quartics. In this thesis, we apply the Gram matrix method to construct and examine cones between the two cones along projective varieties containing the Veronese variety. Introducing a specific filtration of projective varieties containing the Veronese variety, we provide a specific cone filtration. In a non-Hilbert case, at least one of the inclusions in this filtration has to be strict, but it is not clear which one and how many. The identification of each strict inclusion is the goal of this thesis. We show that the first n+1 (respectively n+2 if n=2) cones of the filtration coincide with the cone of sums of squares by leveraging a result by Blekherman-Smith-Velasco. Moreover, we develop two degree-jumping principles to demonstrate that all remaining inclusions are strict. This allows us to give a refinement of Hilbert’s 1888 theorem. Lastly, we apply a method of Scheiderer to show that each identified strictly separating intermediate cone fails to be a spectrahedral shadow. We therefore provide many counterexamples to the Helton-Nie conjecture that any convex semialgebraic set is a spectrahedral shadow.

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Fachgebiet (DDC)
510 Mathematik

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Real algebraic geometry, Positive polynomials, Sums of squares, Varieties of Minimal Degree, Separating Cones, Degree-Jumping Principle, Circuit Polynomials, PSD-Extremality, spectrahedral shadows, Helton-Nie Conjecture, Toric Varieties

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ISO 690HESS, Sarah, 2024. Hilbert’s 1888 Theorem for Cones along the Veronese Variety [Dissertation]. Konstanz: Universität Konstanz
BibTex
@phdthesis{Hess2024Hilbe-71343,
  year={2024},
  title={Hilbert’s 1888 Theorem for Cones along the Veronese Variety},
  author={Hess, Sarah},
  address={Konstanz},
  school={Universität Konstanz}
}
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In this thesis, we apply the Gram matrix method to construct and examine cones between the two cones along projective varieties containing the Veronese variety. Introducing a specific filtration of projective varieties containing the Veronese variety, we provide a specific cone filtration. In a non-Hilbert case, at least one of the inclusions in this filtration has to be strict, but it is not clear which one and how many. The identification of each strict inclusion is the goal of this thesis. 
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Prüfungsdatum der Dissertation

October 23, 2024
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Konstanz, Univ., Diss., 2024
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