Publikation: Hilbert’s 1888 Theorem for Cones along the Veronese Variety
Dateien
Datum
Autor:innen
Herausgeber:innen
ISSN der Zeitschrift
Electronic ISSN
ISBN
Bibliografische Daten
Verlag
Schriftenreihe
Auflagebezeichnung
URI (zitierfähiger Link)
Internationale Patentnummer
Link zur Lizenz
Angaben zur Forschungsförderung
Projekt
Open Access-Veröffentlichung
Sammlungen
Core Facility der Universität Konstanz
Titel in einer weiteren Sprache
Publikationstyp
Publikationsstatus
Erschienen in
Zusammenfassung
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.
Zusammenfassung in einer weiteren Sprache
Fachgebiet (DDC)
Schlagwörter
Konferenz
Rezension
Zitieren
ISO 690
HESS, Sarah, 2024. Hilbert’s 1888 Theorem for Cones along the Veronese Variety [Dissertation]. Konstanz: Universität KonstanzBibTex
@phdthesis{Hess2024Hilbe-71343, year={2024}, title={Hilbert’s 1888 Theorem for Cones along the Veronese Variety}, author={Hess, Sarah}, address={Konstanz}, school={Universität Konstanz} }
RDF
<rdf:RDF xmlns:dcterms="http://purl.org/dc/terms/" xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns:bibo="http://purl.org/ontology/bibo/" xmlns:dspace="http://digital-repositories.org/ontologies/dspace/0.1.0#" xmlns:foaf="http://xmlns.com/foaf/0.1/" xmlns:void="http://rdfs.org/ns/void#" xmlns:xsd="http://www.w3.org/2001/XMLSchema#" > <rdf:Description rdf:about="https://kops.uni-konstanz.de/server/rdf/resource/123456789/71343"> <dc:creator>Hess, Sarah</dc:creator> <dc:language>eng</dc:language> <dcterms:hasPart rdf:resource="https://kops.uni-konstanz.de/bitstream/123456789/71343/4/Hess_2-1akwldzdprsoe2.pdf"/> <dspace:isPartOfCollection rdf:resource="https://kops.uni-konstanz.de/server/rdf/resource/123456789/39"/> <dc:rights>terms-of-use</dc:rights> <dc:contributor>Hess, Sarah</dc:contributor> <bibo:uri rdf:resource="https://kops.uni-konstanz.de/handle/123456789/71343"/> <dcterms:available rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">2024-11-20T07:22:42Z</dcterms:available> <void:sparqlEndpoint rdf:resource="http://localhost/fuseki/dspace/sparql"/> <dcterms:issued>2024</dcterms:issued> <foaf:homepage rdf:resource="http://localhost:8080/"/> <dcterms:abstract>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.</dcterms:abstract> <dcterms:title>Hilbert’s 1888 Theorem for Cones along the Veronese Variety</dcterms:title> <dcterms:rights rdf:resource="https://rightsstatements.org/page/InC/1.0/"/> <dcterms:isPartOf rdf:resource="https://kops.uni-konstanz.de/server/rdf/resource/123456789/39"/> <dc:date rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">2024-11-20T07:22:42Z</dc:date> <dspace:hasBitstream rdf:resource="https://kops.uni-konstanz.de/bitstream/123456789/71343/4/Hess_2-1akwldzdprsoe2.pdf"/> </rdf:Description> </rdf:RDF>