Zero-set property of $o$-minimal indefinitely Peano differentiable functions

Andreas Fischer¹

¹ Department of Mathematics & Statistics University of Saskatchewan 106 Wiggins Road Saskatoon, SK, S7N 5E6, Canada

Annales Polonici Mathematici, Tome 94 (2008) no. 1, pp. 29-41

Résumé

Given an $o$-minimal expansion $\mathcal M$ of a real closed field $R$ which is not polynomially bounded. Let $\mathcal {TP}^\infty$ denote the definable indefinitely Peano differentiable functions. If we further assume that $\mathcal M$ admits $\mathcal {TP}^{\infty}$ cell decomposition, each definable closed subset $A$ of $R^n$ is the zero-set of a $\mathcal {TP}^{\infty}$ function $f:R^n\rightarrow R$. This implies $\mathcal {TP}^\infty$ approximation of definable continuous functions and gluing of $\mathcal {TP}^\infty$ functions defined on closed definable sets.

Détail

DOI : 10.4064/ap94-1-3

Keywords: given o minimal expansion mathcal real closed field which polynomially bounded mathcal infty denote definable indefinitely peano differentiable functions further assume mathcal admits mathcal infty cell decomposition each definable closed subset zero set mathcal infty function rightarrow implies mathcal infty approximation definable continuous functions gluing mathcal infty functions defined closed definable sets