Extending Tamm's theorem

Lou van den Dries; Chris Miller

Annales de l'institut Fourier (1994)

  • Volume: 44, Issue: 5, page 1367-1395
  • ISSN: 0373-0956

Abstract

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We extend a result of M. Tamm as follows:Let f : A → ℝ , A ⊆ 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 ↦ x r : ( 0 , ∞ ) → ℝ , r ∈ ℝ . Then there exists N ∈ ℕ such that for all ( a , b ) ∈ A , if y ↦ f ( a , y ) is C N in a neighborhood of b , then y ↦ f ( a , y ) is real analytic in a neighborhood of b .

How to cite

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Dries, Lou van den, and Miller, Chris. "Extending Tamm's theorem." Annales de l'institut Fourier 44.5 (1994): 1367-1395. <http://eudml.org/doc/75102>.

@article{Dries1994,
abstract = {We extend a result of M. Tamm as follows:Let $f:A\rightarrow \{\Bbb R\},\, A\subseteq \{\Bbb R\}^\{m+n\}$, be definable in the ordered field of real numbers augmented by all real analytic functions on compact boxes and all power functions $x\mapsto x^r:(0,\infty )\rightarrow \{\Bbb R\},\, r\in \{\Bbb R\}$. Then there exists $N\in \{\Bbb N\}$ 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 $b$.},
author = {Dries, Lou van den, Miller, Chris},
journal = {Annales de l'institut Fourier},
keywords = {o-minimal structure; polynomially bounded structure; Gateaux derivative; real analytic; quasianalytic; finitely subanalytic; Tarski-Seidenberg property; Puiseux expansion},
language = {eng},
number = {5},
pages = {1367-1395},
publisher = {Association des Annales de l'Institut Fourier},
title = {Extending Tamm's theorem},
url = {http://eudml.org/doc/75102},
volume = {44},
year = {1994},
}

TY - JOUR
AU - Dries, Lou van den
AU - Miller, Chris
TI - Extending Tamm's theorem
JO - Annales de l'institut Fourier
PY - 1994
PB - Association des Annales de l'Institut Fourier
VL - 44
IS - 5
SP - 1367
EP - 1395
AB - We extend a result of M. Tamm as follows:Let $f:A\rightarrow {\Bbb R},\, A\subseteq {\Bbb R}^{m+n}$, be definable in the ordered field of real numbers augmented by all real analytic functions on compact boxes and all power functions $x\mapsto x^r:(0,\infty )\rightarrow {\Bbb R},\, r\in {\Bbb R}$. Then there exists $N\in {\Bbb N}$ 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 $b$.
LA - eng
KW - o-minimal structure; polynomially bounded structure; Gateaux derivative; real analytic; quasianalytic; finitely subanalytic; Tarski-Seidenberg property; Puiseux expansion
UR - http://eudml.org/doc/75102
ER -

References

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  1. [AM] S. ABHYANKAR and T. MOH, Reduction theorem for divergent power series, J. Reine Angew. Math., 241 (1970), 27-33. Zbl0191.04403MR41 #3800
  2. [BM] E. BIERSTONE and P. MILMAN, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math., 67 (1988), 5-42. Zbl0674.32002MR89k:32011
  3. [BS] J. BOCHNAK and J. SICIAK, Analytic functions in topological vector spaces, Studia Math., 39 (1971), 77-112. Zbl0214.37703MR47 #2365
  4. [DD] J. DENEF and L. van den DRIES, p-adic and real subanalytic sets, Ann. of Math., 128 (1988), 79-138. Zbl0693.14012MR89k:03034
  5. [D1] L. van den DRIES, Remarks on Tarski's problem concerning (ℝ, +, ., exp), Logic Colloquium 1982, eds. G. Lolli, G. Longo and A. Marcja, North Holland, Amsterdam (1984), 97-121. Zbl0585.03006MR86g:03052
  6. [D2] L. van den DRIES, A generalization of the Tarski-Seidenberg theorem and some nondefinability results, Bull. Amer. Math. Soc. (N.S.), 15 (1986), 189-193. Zbl0612.03008MR88b:03048
  7. [D3] L. van den DRIES, Tame topology and o-minimal structures, (monograph in preparation). Zbl0953.03045
  8. [DMM] L. van den DRIES, A. MACINTYRE and D. MARKER, The elementary theory of restricted analytic fields with exponentiation, Ann. of Math., 140 (1994), 183-205. Zbl0837.12006MR95k:12015
  9. [DM] L. van den DRIES and C. MILLER, On the real exponential field with restricted analytic functions, Israel J. Math., 85 (1994), 19-56. Zbl0823.03017MR95e:03099
  10. [H] R. HARDT, Semi-algebraic local triviality in semi-algebraic mappings, Amer. J. Math., 102 (1980), 291-302. Zbl0465.14012MR81d:32012
  11. [KPS] J. KNIGHT, A. PILLAY and C. STEINHORN, Definable sets in ordered structures. II, Trans. Amer. Math. Soc., 295 (1986), 593-605. Zbl0662.03024MR88b:03050b
  12. [KR] T. KAYAL and G. RABY, Ensembles sous-analytiques: quelques propriétés globales, C. R. Acad. Sci. Paris, Sér. I Math., 308 (1989), 521-523. Zbl0674.32003MR90d:32016
  13. [K] K. KURDYKA, Points réguliers d'un sous-analytique, Ann. Inst. Fourier (Grenoble), 38-1 (1988), 133-156. Zbl0619.32007MR89g:32010
  14. [M1] C. MILLER, Exponentiation is hard to avoid, Proc. Amer. Math. Soc., 122 (1994), 257-259. Zbl0808.03022MR94k:03042
  15. [M2] C. MILLER, Expansions of the real field with power functions, Ann. Pure Appl. Logic, 68 (1994), 79-94. Zbl0823.03018MR95i:03081
  16. [M3] C. MILLER, Infinite differentiability in polynomially bounded o-minimal structures, Proc. Amer. Math. Soc., (to appear). Zbl0823.03019
  17. [P] W. PAWLUCKI, Le théorème de Puiseux pour une application sous-analytique, Bull. Polish Acad. Sci. Math., 32 (1984), 556-560. Zbl0574.32010MR86j:32015
  18. [PS] A. PILLAY and C. STEINHORN, Definable sets in ordered structures. I, Trans. Amer. Math. Soc., 295 (1986), 565-592. Zbl0662.03023MR88b:03050a
  19. [T] M. TAMM, Subanalytic sets in the calculus of variations, Acta Math., 146 (1981), 167-199. Zbl0478.58010MR82h:32012
  20. [To] J.-Cl. TOUGERON, Algèbres analytiques topologiquement noethériennes. Théorie de Khovanskii, Ann. Inst. Fourier (Grenoble), 41-4 (1991), 823-840. Zbl0786.32011MR93f:32005
  21. [W] A. WILKIE, Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, Jour. Amer. Math. Soc., (to appear). Zbl0892.03013

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