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dc.contributor.authorBasu, Saugata
dc.contributor.authorRiener, Cordian
dc.date.accessioned2019-03-19T13:01:19Z
dc.date.available2019-03-19T13:01:19Z
dc.date.issued2018-03-02
dc.description.abstractLet R be a real closed field. We prove that for any fixed <i>d</i>, the equivariant rational cohomology groups of closed symmetric semi-algebraic subsets of R<sup><i>k</i></sup> defined by polynomials of degrees bounded by <i>d</i> vanishes in dimensions <i>d</i> and larger. This vanishing result is tight. Using a new geometric approach we also prove an upper bound of [mathematical formula] on the equivariant Betti numbers of closed symmetric semi-algebraic subsets of R<sup><i>k</i></sup> defined by quantifier-free formulas involving <i>s</i> symmetric polynomials of degrees bounded by <i>d</i>, where 1<<i>d</i>≪<i>s,k</i>. This bound is tight up to a factor depending only on <i>d</i>. These results significantly improve upon those obtained previously in Basu and Riener (Adv Math 305:803–855, 2017) which were proved using different techniques. Our new methods are quite general, and also yield bounds on the equivariant Betti numbers of certain special classes of symmetric definable sets (definable sets symmetrized by pulling back under symmetric polynomial maps of fixed degree) in arbitrary o-minimal structures over R. Finally, we utilize our new approach to obtain an algorithm with polynomially bounded complexity for computing these equivariant Betti numbers. In contrast, the problem of computing the ordinary Betti numbers of (not necessarily symmetric) semi-algebraic sets is considered to be an intractable problem, and all known algorithms for this problem have doubly exponential complexity.en_US
dc.description.sponsorshipUS National Science Foundation Tromsø forskningsstiftelseen_US
dc.descriptionThis is a post-peer-review, pre-copyedit version of an article published in <i>Selecta Mathematica, New Series</i>. The final authenticated version is available online at: <a href=https://doi.org/10.1007/s00029-018-0401-7>https://doi.org/10.1007/s00029-018-0401-7</a>.en_US
dc.identifier.citationBasu, S. & Riener, C. (2018). On the equivariant Betti numbers of symmetric definable sets: vanishing, bounds and algorithms. <i>Selecta Mathematica, New Series, 24</i>(4), 3241-3281. https://doi.org/10.1007/s00029-018-0401-7en_US
dc.identifier.cristinIDFRIDAID 1573463
dc.identifier.doi10.1007/s00029-018-0401-7
dc.identifier.issn1022-1824
dc.identifier.issn1420-9020
dc.identifier.urihttps://hdl.handle.net/10037/15024
dc.language.isoengen_US
dc.publisherSpringeren_US
dc.relation.journalSelecta Mathematica, New Series
dc.rights.accessRightsopenAccessen_US
dc.subjectVDP::Mathematics and natural science: 400::Mathematics: 410::Algebra/algebraic analysis: 414en_US
dc.subjectVDP::Matematikk og Naturvitenskap: 400::Matematikk: 410::Algebra/algebraisk analyse: 414en_US
dc.subjectEquivariant cohomologyen_US
dc.subjectSymmetric semi-algebraic setsen_US
dc.subjectBetti numbersen_US
dc.subjectComputational complexityen_US
dc.titleOn the equivariant Betti numbers of symmetric definable sets: vanishing, bounds and algorithmsen_US
dc.typeJournal articleen_US
dc.typeTidsskriftartikkelen_US
dc.typePeer revieweden_US


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