Rigid subanalytic sets

2002 ◽  
pp. 141-150
Author(s):  
T. Gardener ◽  
Hans Schoutens
Keyword(s):  
Subanalytic Sets

scholarly journals Arc-smooth functions on closed sets

2019 ◽  
Vol 155 (4) ◽  
pp. 645-680 ◽  
Author(s):  
Armin Rainer
Keyword(s):  
Holomorphic Function ◽  
Smooth Function ◽  
Smooth Curve ◽  
Smooth Functions ◽  
Closed Sets ◽  
Open Set ◽  
Connected Set ◽  
Subanalytic Sets ◽  
Real Analytic ◽  
Analytic Curves

By an influential theorem of Boman, a function $f$ on an open set $U$ in $\mathbb{R}^{d}$ is smooth (${\mathcal{C}}^{\infty }$) if and only if it is arc-smooth, that is, $f\,\circ \,c$ is smooth for every smooth curve $c:\mathbb{R}\rightarrow U$. In this paper we investigate the validity of this result on closed sets. Our main focus is on sets which are the closure of their interior, so-called fat sets. We obtain an analogue of Boman’s theorem on fat closed sets with Hölder boundary and on fat closed subanalytic sets with the property that every boundary point has a basis of neighborhoods each of which intersects the interior in a connected set. If $X\subseteq \mathbb{R}^{d}$ is any such set and $f:X\rightarrow \mathbb{R}$ is arc-smooth, then $f$ extends to a smooth function defined on $\mathbb{R}^{d}$. We also get a version of the Bochnak–Siciak theorem on all closed fat subanalytic sets and all closed sets with Hölder boundary: if $f:X\rightarrow \mathbb{R}$ is the restriction of a smooth function on $\mathbb{R}^{d}$ which is real analytic along all real analytic curves in $X$, then $f$ extends to a holomorphic function on a neighborhood of $X$ in $\mathbb{C}^{d}$. Similar results hold for non-quasianalytic Denjoy–Carleman classes (of Roumieu type). We will also discuss sharpness and applications of these results.

scholarly journals Rigid Subanalytic Sets in the Plane

1994 ◽  
Vol 170 (1) ◽  
pp. 266-276 ◽  
Author(s):  
H. Schoutens
Keyword(s):  
Subanalytic Sets

Characterization of the Clarke regularity of subanalytic sets

2017 ◽  
Vol 146 (4) ◽  
pp. 1639-1649
Author(s):  
Abderrahim Jourani ◽  
Moustapha Séne
Keyword(s):  
Subanalytic Sets ◽  
Clarke Regularity

scholarly journals NON-ARCHIMEDEAN YOMDIN–GROMOV PARAMETRIZATIONS AND POINTS OF BOUNDED HEIGHT

2015 ◽  
Vol 3 ◽  
Author(s):  
RAF CLUCKERS ◽  
GEORGES COMTE ◽  
FRANÇOIS LOESER
Keyword(s):  
Algebraic Variety ◽  
Lipschitz Continuity ◽  
General Context ◽  
Complex Polynomials ◽  
Bounded Degree ◽  
Subanalytic Sets ◽  
Definable Sets ◽  
Taylor Polynomials ◽  
Definable Functions ◽  
Totally Disconnected

We prove an analog of the Yomdin–Gromov lemma for $p$-adic definable sets and more broadly in a non-Archimedean definable context. This analog keeps track of piecewise approximation by Taylor polynomials, a nontrivial aspect in the totally disconnected case. We apply this result to bound the number of rational points of bounded height on the transcendental part of $p$-adic subanalytic sets, and to bound the dimension of the set of complex polynomials of bounded degree lying on an algebraic variety defined over $\mathbb{C}(\!(t)\!)$, in analogy to results by Pila and Wilkie, and by Bombieri and Pila, respectively. Along the way we prove, for definable functions in a general context of non-Archimedean geometry, that local Lipschitz continuity implies piecewise global Lipschitz continuity.

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