Geodesic
Fields with analytic structure
Journal of the European Mathematical Society, Tome 13 (2011) no. 4, pp. 1147-1223 Cet article a éte moissonné depuis la source EMS Press

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We present a unifying theory of fields with certain classes of analytic functions, called fields with analytic structure. Both real closed fields and Henselian valued fields are considered. For real closed fields with analytic structure, o-minimality is shown. For Henselian valued fields, both the model theory and the analytic theory are developed. We give a list of examples that comprises, to our knowledge, all principal, previously studied, analytic structures on Henselian valued fields, as well as new ones. The b-minimality is shown, as well as other properties useful for motivic integration on valued fields. The paper is reminiscent of [Denef, van den Dries, p-adic and real subanalytic sets. Ann. of Math. (2) 128 (1988) 79–138], of [Cohen, Paul J. Decision procedures for real and p-adic fields. Comm. Pure Appl. Math. 22 (1969)131–151], and of [Fresnel, van der Put, Rigid analytic geometry and its applications. Progress in Mathematics, 218 Birkhäuser (2004)], and unifies work by van den Dries, Haskell, Macintyre, Macpherson, Marker, Robinson, and the authors.
DOI : 10.4171/jems/278
Classification : 32-XX, 03-XX, 14-XX, 28-XX
Keywords: Henselian valued fields, o-minimality, b-minimality, subanalytic functions, cell decomposition, analytic structure, separated power series. 
@article{JEMS_2011_13_4_a10,
     author = {Raf Cluckers and Leonard Lipshitz},
     title = {Fields with analytic structure},
     journal = {Journal of the European Mathematical Society},
     pages = {1147--1223},
     year = {2011},
     volume = {13},
     number = {4},
     doi = {10.4171/jems/278},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/278/}
}
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Raf Cluckers; Leonard Lipshitz. Fields with analytic structure. Journal of the European Mathematical Society, Tome 13 (2011) no. 4, pp. 1147-1223. doi: 10.4171/jems/278

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