On the set of local extrema of a subanalytic function

• Autores: José Francisco Fernando Galván
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 71, Fasc. 1, 2020, págs. 1-24
• Idioma: inglés
• DOI: 10.1007/s13348-019-00245-6
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• Resumen

  • Let ? 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 ?(?) the family of the subsets of M that belong to the category ?. Let ?:?→ℝ be a subanalytic function on a subset ?∈?(?) such that the inverse image under f of each interval of ℝ belongs to ?(?). Let Max(?) be the set of local maxima of f and consider its level sets Max?(?):=Max(?)∩{?=?}={?=?}∖Cl({?>?}) for each ?∈ℝ. In this work we show that if f is continuous, then Max(?)=⨆?∈ℝMax?(?)∈?(?) if and only if the family {Max?(?)}?∈ℝ is locally finite in M. If we erase continuity condition, there exist subanalytic functions ?:?→? such that Max(?)∈?(?), but the family {Max?(?)}?∈ℝ is not locally finite in M or such that Max(?) is connected but it is not even subanalytic. We show in addition that if ? is the category of subanalytic sets and ?:?→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(?)∈?(?) and the family {Max?(?)}?∈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 ? is either the category of semianalytic sets or the category of C-semianalytic sets and f is the restriction to an ?-subset of M of an analytic function on M, then the family {Max?(?)}?∈ℝ is locally finite in M and Max(?)=⨆?∈ℝMax?(?)∈?(?). We also show that if the category ? contains the intersections of algebraic sets with real analytic submanifolds and ?∈?(?) is not closed in M, then there exists a continuous subanalytic function ?:?→ℝ with graph belonging to ?(?×ℝ) such that inverse images under f of the intervals of ℝ belong to ?(?) but Max(?) does not belong to ?(?). As subanalytic sets are locally connected, the set of non-openness points of a continuous subanalytic function ?:?→ℝ coincides with the set of local extrema Extr(?):=Max(?)∪Min(?). This means that if ?:?→ℝ is a continuous subanalytic function defined on a closed set ?∈?(?) such that the inverse image under f of each interval of ℝ belongs to ?(?), then the set Op(?) of openness points of f belongs to ?(?). Again the closedness of X in M is crucial to guarantee that Op(?) belongs to ?(?). The type of results stated above are straightforward if ? is an o-minimal structure of subanalytic sets. However, the proof of the previous results requires further work for a category ? of subanalytic sets that does not constitute an o-minimal structure.