Differentiable approximation of continuous semialgebraic maps
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José F. Fernando [1] ; Riccardo Ghiloni [2]
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[1] Universidad Complutense de Madrid
Universidad Complutense de Madrid
Madrid, España
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[2] Università di Trento, Italia
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- Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 25, Nº. 3, 2019
- Idioma: inglés
- DOI: 10.1007/s00029-019-0496-5
- Enlaces
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Resumen
In this work we approach the problem of approximating uniformly continuous semialgebraic maps f:S→T from a compact semialgebraic set S to an arbitrary semialgebraic set T by semialgebraic maps g:S→T that are differentiable of class Cν for a fixed integer ν≥1 . As the reader can expect, the difficulty arises mainly when one tries to keep the same target space after approximation. For ν=1 we give a complete affirmative solution to the problem: such a uniform approximation is always possible. For ν≥2 we obtain density results in the following two relevant situations: either T is compact and locally Cν semialgebraically equivalent to a polyhedron, for instance when T is a compact polyhedron; or T is an open semialgebraic subset of a Nash set, for instance when T is a Nash set. Our density results are based on a recent C1 -triangulation theorem for semialgebraic sets due to Ohmoto and Shiota, and on new approximation techniques we develop in the present paper. Our results are sharp in a sense we specify by explicit examples.
