1. Strict Cp-triangulations -- a new approach to desingularization.
- Author
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Pawłucki, Wiesław
- Subjects
- *
SEMIALGEBRAIC sets , *TRIANGULATION , *DETECTORS , *SET-valued maps , *INTEGERS - Abstract
Let R be any real closed field expanded by some o-minimal structure. Let f : A → Rd be a definable and continuous mapping defined on a definable, closed, bounded subset A of Rn. Let E be a finite family of definable subsets of Rn contained in A. Let p be any positive integer. We prove that then there exists a finite simplicial complex T in Rn and a definable homeomorphism h :|T| → A, where |T| := υT, such that for each simplex Δ ∈ T, the restriction of h to its relative interior ... is a Cp-embedding of ... into Rn and moreover both h and f o h are of class Cp in the sense that they have definable Cp-extensions defined on an open definable neighborhood of |T| in Rn. We then call a pair (T, h) a strict Cp-triangulation of A. In addition, this triangulation can be made compatible with E in the sense that for each E ∈ E, h-1(E) is a union of some ... where Δ ∈ T. We also give an application to approximation theory. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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