## Publication: On the set of local extrema of a subanalytic function

 dc.contributor.author Fernando Galván, José Francisco dc.date.accessioned 2023-06-17T08:30:20Z dc.date.available 2023-06-17T08:30:20Z dc.date.issued 2020 dc.description.abstract Let F be a category of subanalytic subsets of real analytic manifolds that is closed under basic set-theoretical operations (locally finite unions, difference and product) and basic topological operations (taking connected components and closures). Let M be a real analytic manifold and denote F(M) the family of the subsets of M that belong to the category F. Let f:X→R be a subanalytic function on a subset X∈F(M) such that the inverse image under f of each interval of R belongs to F(M). Let Max(f) be the set of local maxima of f and consider its level sets Maxλ(f):=Max(f)∩{f=λ}={f=λ}∖Cl({f>λ}) for each λ∈R. In this work we show that if f is continuous, then Max(f)=⨆λ∈RMaxλ(f)∈F(M) if and only if the family {Maxλ(f)}λ∈R is locally finite in M. If we erase continuity condition, there exist subanalytic functions f:X→M such that Max(f)∈F(M), but the family {Maxλ(f)}λ∈R is not locally finite in M or such that Max(f) is connected but it is not even subanalytic. We show in addition that if F is the category of subanalytic sets and f:X→R is a (non-necessarily continuous) subanalytic map f that maps relatively compact subsets of M contained in X to bounded subsets of R, then Max(f)∈F(M) and the family {Maxλ(f)}λ∈R is locally finite in M. An example of this type of functions are continuous subanalytic functions on closed subanalytic subsets of M. The previous results imply that if F is either the category of semianalytic sets or the category of C-semianalytic sets and f is the restriction to an F-subset of M of an analytic function on M, then the family {Maxλ(f)}λ∈R is locally finite in M and Max(f)=⨆λ∈RMaxλ(f)∈F(M). We also show that if the category F contains the intersections of algebraic sets with real analytic submanifolds and X∈F(M) is not closed in M, then there exists a continuous subanalytic function f:X→R with graph belonging to F(M×R) such that inverse images under f of the intervals of R belong to F(M) but Max(f) does not belong to F(M). As subanalytic sets are locally connected, the set of non-openness points of a continuous subanalytic function f:X→R coincides with the set of local extrema Extr(f):=Max(f)∪Min(f). This means that if f:X→R is a continuous subanalytic function defined on a closed set X∈F(M) such that the inverse image under f of each interval of R belongs to F(M), then the set Op(f) of openness points of f belongs to F(M). Again the closedness of X in M is crucial to guarantee that Op(f) belongs to F(M). The type of results stated above are straightforward if F is an o-minimal structure of subanalytic sets. However, the proof of the previous results requires further work for a category F of subanalytic sets that does not constitute an o-minimal structure. dc.description.department Depto. de Álgebra, Geometría y Topología dc.description.faculty Fac. de Ciencias Matemáticas dc.description.refereed FALSE dc.description.sponsorship Ministerio de Economía y Competitividad (MINECO) dc.description.sponsorship Universidad Complutense de Madrid dc.description.status pub dc.eprint.id https://eprints.ucm.es/id/eprint/75947 dc.identifier.doi 10.1007/s13348-019-00245-6 dc.identifier.issn 0010-0757 dc.identifier.officialurl https://doi.org/10.1007/s13348-019-00245-6 dc.identifier.uri https://hdl.handle.net/20.500.14352/7313 dc.journal.title Collectanea Mathematica dc.language.iso eng dc.page.final 24 dc.page.initial 1 dc.publisher Springer dc.relation.projectID GRAYAS (MTM2014-55565-P) dc.relation.projectID UCM (910444) dc.rights.accessRights open access dc.subject.cdu 512.7 dc.subject.keyword Subanalytic set dc.subject.keyword Semianalytic set dc.subject.keyword C-seminalytic set dc.subject.keyword Weak category dc.subject.keyword Subanalytic function dc.subject.keyword Semianalytic function dc.subject.keyword C-semianalytic function dc.subject.keyword Analytic function dc.subject.keyword Locally normal crossing analytic function dc.subject.keyword Local maxima dc.subject.keyword Local minima dc.subject.keyword Local extrema dc.subject.ucm Geometria algebraica dc.subject.unesco 1201.01 Geometría Algebraica dc.title On the set of local extrema of a subanalytic function dc.type journal article dc.volume.number 71 dspace.entity.type Publication relation.isAuthorOfPublication 499732d5-c130-4ea6-8541-c4ec934da408 relation.isAuthorOfPublication.latestForDiscovery 499732d5-c130-4ea6-8541-c4ec934da408
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