Sums of hermitian squares and the BMV conjecture

Klep, Igor; Schweighofer, Markus

Sums of hermitian squares and the BMV conjecture

Dateien

bmv.pdfGröße: 1.64 MBDownloads: 223

Datum

2008

Open Access-Veröffentlichung

Open Access Green

Sammlungen

Mathematik und Statistik

Publikationstyp

Zeitschriftenartikel

Publikationsstatus

Published

Erschienen in

Journal of Statistical Physics. 2008, 133(4), pp. 739-760. ISSN 0022-4715. Available under: doi: 10.1007/s10955-008-9632-x

Zusammenfassung

Abstract. We show that all the coe cients of the polynomial tr((A + tB)^m) ∈ ℝ[t] are nonnegative whenever m ≤ 13 is a nonnegative integer and A and B are positive semide nite matrices of the same size. This has previously been known only for m ≤ 7. The validity of the statement for arbitrary m has recently been shown to be equivalent to the Bessis-Moussa-Villani conjecture from theoretical physics. In our proof, we establish a connection to sums of hermitian squares of polynomials in noncommuting variables and to semide nite programming. As a by-product we obtain an example of a real polynomial in two noncommuting variables having nonnegative trace on all symmetric matrices of the same size, yet not being a sum of hermitian squares and commutators.

Fachgebiet (DDC)

510 Mathematik

Schlagwörter

Bessis-Moussa-Villani (BMV) conjecture, sum of hermitian squares, trace inequality, semide nite programming.

Zitieren

ISO 690

KLEP, Igor, Markus SCHWEIGHOFER, 2008. Sums of hermitian squares and the BMV conjecture. In: Journal of Statistical Physics. 2008, 133(4), pp. 739-760. ISSN 0022-4715. Available under: doi: 10.1007/s10955-008-9632-x

BibTex

@article{Klep2008hermi-15620,
  year={2008},
  doi={10.1007/s10955-008-9632-x},
  title={Sums of hermitian squares and the BMV conjecture},
  number={4},
  volume={133},
  issn={0022-4715},
  journal={Journal of Statistical Physics},
  pages={739--760},
  author={Klep, Igor and Schweighofer, Markus}
}

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/123456789/15620">
    <void:sparqlEndpoint rdf:resource="http://localhost/fuseki/dspace/sparql"/>
    <dspace:hasBitstream rdf:resource="https://kops.uni-konstanz.de/bitstream/123456789/15620/1/bmv.pdf"/>
    <dspace:isPartOfCollection rdf:resource="https://kops.uni-konstanz.de/server/rdf/resource/123456789/39"/>
    <dc:date rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">2011-10-31T09:23:01Z</dc:date>
    <dc:rights>terms-of-use</dc:rights>
    <dcterms:issued>2008</dcterms:issued>
    <dc:language>eng</dc:language>
    <foaf:homepage rdf:resource="http://localhost:8080/"/>
    <dc:contributor>Klep, Igor</dc:contributor>
    <dc:creator>Schweighofer, Markus</dc:creator>
    <dcterms:title>Sums of hermitian squares and the BMV conjecture</dcterms:title>
    <dcterms:available rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">2011-10-31T09:23:01Z</dcterms:available>
    <dc:creator>Klep, Igor</dc:creator>
    <dcterms:hasPart rdf:resource="https://kops.uni-konstanz.de/bitstream/123456789/15620/1/bmv.pdf"/>
    <dc:contributor>Schweighofer, Markus</dc:contributor>
    <dcterms:isPartOf rdf:resource="https://kops.uni-konstanz.de/server/rdf/resource/123456789/39"/>
    <bibo:uri rdf:resource="http://kops.uni-konstanz.de/handle/123456789/15620"/>
    <dcterms:bibliographicCitation>First publ. in: Journal of Statistical Physics 133 (2008), 4. - S. 739-760</dcterms:bibliographicCitation>
    <dcterms:rights rdf:resource="https://rightsstatements.org/page/InC/1.0/"/>
    <dcterms:abstract xml:lang="eng">Abstract. We show that all the coe cients of the polynomial tr((A + tB)&lt;sup&gt;m&lt;/sup&gt;)	&amp;#8712; ℝ[t] are nonnegative whenever m &amp;#8804; 13 is a nonnegative integer and A and B are positive semide nite matrices of the same size. This has previously been known only for m &amp;#8804; 7. The validity of the statement for arbitrary m has recently been shown to be equivalent to the Bessis-Moussa-Villani conjecture from theoretical physics. In our proof, we establish a connection to sums of hermitian squares of polynomials in noncommuting variables and to semide nite programming. As a by-product we obtain an example of a real polynomial in two noncommuting variables having nonnegative trace on all symmetric matrices of the same size, yet not being a sum of hermitian squares and commutators.</dcterms:abstract>
  </rdf:Description>
</rdf:RDF>

Universitätsbibliographie

Ja