Extending Tamm's theorem

Dries, Lou van den; Miller, Chris

doi:10.5802/aif.1438

Dries, Lou van den ; Miller, Chris

Annales de l'Institut Fourier, Tome 44 (1994) no. 5, pp. 1367-1395

Abstract (VO)
Résumé

We extend a result of M. Tamm as follows:

Let $f : A \to ℝ, A \subseteq ℝ^{m + n}$ , be definable in the ordered field of real numbers augmented by all real analytic functions on compact boxes and all power functions $x \mapsto x^{r} : (0, \infty) \to ℝ, r \in ℝ$ . Then there exists $N \in ℕ$ such that for all $(a, b) \in A$ , if $y \mapsto f (a, y)$ is $C^{N}$ in a neighborhood of $b$ , then $y \mapsto f (a, y)$ is real analytic in a neighborhood of $b$ .

On généralise un résultat de M. Tamm :

Soit $f : A \to ℝ$ , $A \subseteq ℝ^{m + n}$ , définissable dans le corps ordonné des nombres réels augmenté par toutes les fonctions analytiques réelles sur les cubes compacts et toutes les puissances $x \mapsto x^{r} : (0, \infty) \to ℝ$ , $r \in ℝ$ . Alors, il existe $N \in ℕ$ telle que pour chaque $(a, b) \in A$ , la fonction $y \mapsto f (a, y)$ est $C^{N}$ dans un voisinage de $b$ si et seulement si $y \mapsto f (a, y)$ est analytique dans un voisinage de $b$ .

MR Zbl

DOI : 10.5802/aif.1438

@article{AIF_1994__44_5_1367_0,
     author = {Dries, Lou van den and Miller, Chris},
     title = {Extending {Tamm's} theorem},
     journal = {Annales de l'Institut Fourier},
     pages = {1367--1395},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {44},
     number = {5},
     year = {1994},
     doi = {10.5802/aif.1438},
     zbl = {0816.32004},
     mrnumber = {96g:32016},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1438/}
}

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%A Dries, Lou van den
%A Miller, Chris
%T Extending Tamm's theorem
%J Annales de l'Institut Fourier
%D 1994
%P 1367-1395
%V 44
%N 5
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1438/
%R 10.5802/aif.1438
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Dries, Lou van den; Miller, Chris. Extending Tamm's theorem. Annales de l'Institut Fourier, Tome 44 (1994) no. 5, pp. 1367-1395. doi: 10.5802/aif.1438

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