Type of Publication:  Dissertation 
URI (citable link):  http://nbnresolving.de/urn:nbn:de:bsz:3520263940 
Author:  Goel, Charu 
Year of publication:  2014 
Summary: 
We compare the cone of positive semidefinite (real) forms to its subcone of sum of squares of (real) forms under the additional assumption of symmetry on the given forms. The aim was to generalize a classical theorem of Hilbert from 1888, namely, a positive semidefinite form (psd) in n variables and of degree 2d is a sum of squares (sos) if and only if n=2 or d=1 or (n,2d)=(3,4); for symmetric and even symmetric forms respectively. As main results we construct explicitly psd not sos symmetric quartic forms in more than 4 variables, thereby completing the analogue of Hilbert's 1888 theorem for symmetric forms, which was asserted by Choi and Lam in 1976. Moreover, we construct psd not sos even symmetric octic forms in more than 4 variables and introduce a degree jumping principle to increase the degree of a psd not sos even symmetric form while simultaneously preserving the psd not sos even symmetric property. Finally using these constructions and techniques we present a version of Hilbert's 1888 theorem for even symmetric forms.

Examination date (for dissertations):  Aug 1, 2014 
Dissertation note:  Doctoral dissertation, University of Konstanz 
Subject (DDC):  510 Mathematics 
Keywords:  Real algebraic geometry, Positive polynomials, Sums of squares, Symmmetric forms, Even symmetric forms, Degree jumping principle 
Link to License:  In Copyright 
Bibliography of Konstanz:  Yes 
GOEL, Charu, 2014. Extension of Hilbert's 1888 Theorem to Even Symmetric Forms [Dissertation]. Konstanz: University of Konstanz
@phdthesis{Goel2014Exten29352, title={Extension of Hilbert's 1888 Theorem to Even Symmetric Forms}, year={2014}, author={Goel, Charu}, address={Konstanz}, school={Universität Konstanz} }
<rdf:RDF xmlns:dcterms="http://purl.org/dc/terms/" xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:rdf="http://www.w3.org/1999/02/22rdfsyntaxns#" xmlns:bibo="http://purl.org/ontology/bibo/" xmlns:dspace="http://digitalrepositories.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.unikonstanz.de/rdf/resource/123456789/29352"> <dcterms:rights rdf:resource="https://rightsstatements.org/page/InC/1.0/"/> <dc:date rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">20141127T15:25:08Z</dc:date> <dspace:isPartOfCollection rdf:resource="https://kops.unikonstanz.de/rdf/resource/123456789/39"/> <dcterms:isPartOf rdf:resource="https://kops.unikonstanz.de/rdf/resource/123456789/39"/> <bibo:uri rdf:resource="http://kops.unikonstanz.de/handle/123456789/29352"/> <dc:rights>termsofuse</dc:rights> <dcterms:available rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">20141127T15:25:08Z</dcterms:available> <dcterms:abstract xml:lang="eng">We compare the cone of positive semidefinite (real) forms to its subcone of sum of squares of (real) forms under the additional assumption of symmetry on the given forms. The aim was to generalize a classical theorem of Hilbert from 1888, namely, a positive semidefinite form (psd) in n variables and of degree 2d is a sum of squares (sos) if and only if n=2 or d=1 or (n,2d)=(3,4); for symmetric and even symmetric forms respectively. As main results we construct explicitly psd not sos symmetric quartic forms in more than 4 variables, thereby completing the analogue of Hilbert's 1888 theorem for symmetric forms, which was asserted by Choi and Lam in 1976. Moreover, we construct psd not sos even symmetric octic forms in more than 4 variables and introduce a degree jumping principle to increase the degree of a psd not sos even symmetric form while simultaneously preserving the psd not sos even symmetric property. Finally using these constructions and techniques we present a version of Hilbert's 1888 theorem for even symmetric forms.</dcterms:abstract> <dspace:hasBitstream rdf:resource="https://kops.unikonstanz.de/bitstream/123456789/29352/3/Goel_0263940.pdf"/> <dc:creator>Goel, Charu</dc:creator> <foaf:homepage rdf:resource="http://localhost:8080/jspui"/> <dcterms:hasPart rdf:resource="https://kops.unikonstanz.de/bitstream/123456789/29352/3/Goel_0263940.pdf"/> <dc:language>eng</dc:language> <void:sparqlEndpoint rdf:resource="http://localhost/fuseki/dspace/sparql"/> <dcterms:title>Extension of Hilbert's 1888 Theorem to Even Symmetric Forms</dcterms:title> <dc:contributor>Goel, Charu</dc:contributor> <dcterms:issued>2014</dcterms:issued> </rdf:Description> </rdf:RDF>
Goel_0263940.pdf  1457 