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On semi-analytic sets with Gevrey conditions at the boundary. (Sur les ensembles semi-analytiques avec conditions Gevrey du bord.) (French) Zbl 0803.32003
This is one of a series of works by an outstanding mathematician on a very interesting topic, having its roots in the works of A. S. Khovanskij [Funct. Anal. Appl. 18, 119-127 (1984); translation from Funkts. Anal. Prilozh. 18, No. 2, 40-50 (1984; Zbl 0784.32016)], closely related to the problems of the first return map [B. Malgrange and J.-P. Ramis, Ann. Inst. Fourier 42, No. 1-2, 353-368 (1992; Zbl 0759.34007)] or Pfaffian varieties [R. Moussu and C. Roche, Invent. Math. 105, No. 2, 431-441 (1991; Zbl 0769.58050)].
Very roughly speaking one wants to obtain a class of functions larger than analytic (subanalytic), but still verifying good topological and metric properties. This work is, of course, completely original and different in the approach, the methods and the results from the papers, cited above. Only their roots are similar. The reviewer feels that the best summary is that of the author himself because the work is redactionally perfect, but difficult. The only remark: the semianalytic sets here are NOT the same as the classical semianalytic sets of Łojasiewicz.
Author’s summary: “We consider a subalgebra \(\widetilde G^ \mathbb{R}_ n\) of the algebra of germs at 0 of real \(C^ \infty\) functions on \((\mathbb{R}^ +)^ n\); these germs are analytic on the open quadrant \((\mathbb{R}^{+v})^ n\) and extend to holomorphic functions in some open sets of \(\mathbb{C}^ n\), with Gevrey conditions near the origin. The properties of this algebra are studied. Then, we consider a germ of “semianalytic” set \(X\) at the origin which is defined by a finite number of functions of \(\widetilde G^ \mathbb{R}_ n\), and we prove for \(X\) the same results as for the usual analytic situation: \(X\) admits locally a finite number of connected components; if \(X\neq 0\), \(X\) contains a small Gevrey arc; every function of \(\widetilde G^ \mathbb{R}_ n\) satisfies a Łojasiewicz inequality. At last, we can mix these results with Khovanskii’s theory to get a large class of analytic algebra which verify good topological or metric properties.

MSC:
32B20 Semi-analytic sets, subanalytic sets, and generalizations
32B05 Analytic algebras and generalizations, preparation theorems
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References:
[1] W. BALSER , Summation of Formal Power Series Through Iterated Laplace Transform , Universität Ulm, preliminary version, 1991 . · Zbl 0769.34004
[2] E. BIERSTONE et P. MILMAN , Semi-Analytic and Subanalytic Sets (Publications de l’I.H.E.S., n^\circ 67, 1988 ). Numdam | MR 89k:32011 | Zbl 0674.32002 · Zbl 0674.32002
[3] J. BOCHNAK , M.-F. COSTE-ROY et M. COSTE , Géométrie algébrique réelle , Springer Verlag, Ergebnisse der Mathematik, 1987 . Zbl 0633.14016 · Zbl 0633.14016
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[6] B. MALGRANGE et J.-P. RAMIS , Fonctions multisommables (Annales Institut Fourier, vol. 41, n^\circ 3, 1991 , p. 1-16). Numdam · Zbl 0759.34007
[7] R. MOUSSU et C. ROCHE , Théorie de Khovanskii et problème de Dulac (Inventiones, vol. 105, 1991 , p. 431-441). MR 92e:58169 | Zbl 0769.58050 · Zbl 0769.58050
[8] M. NAGATA , Local Rings (Wiley 1962 , Interscience tracts in pure and applied mathematics, 13). MR 27 #5790 | Zbl 0123.03402 · Zbl 0123.03402
[9] J.-CL. TOUGERON , Algèbres analytiques topologiquement noethériennes (théorie de Khovanskii) (Annales Institut Fourier, vol. 41, n^\circ 4, 1991 ). Numdam | MR 93f:32005 | Zbl 0786.32011 · Zbl 0786.32011
[10] J.-CL. TOUGERON , Inégalités de Lojasiewicz globales (Annales de l’Institut Fourier, vol. 41, n^\circ 4, 1991 ). Numdam | MR 93f:32006 | Zbl 0748.32007 · Zbl 0748.32007
[11] J.-CL. TOUGERON An Introduction to the Theory of Gevrey Expansions and the Borel Laplace Transform with some Applications (cours de 3e cycle, 1990 ) et Sur les ensembles analytiques réels définis par des équations Gevrey au bord , manuscrit, 1990 .
[12] J.-CL. TOUGERON , Idéaux de fonctions différentiables (Ergebnisse der Mathematik, vol. 71, 1972 ). MR 55 #13472 | Zbl 0251.58001 · Zbl 0251.58001
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