Strictly convergent analytic structures

Raf Cluckers ; Leonard Lipshitz

Journal of the European Mathematical Society, Tome 19 (2017) no. 1, pp. 107-149.

Résumé

We give conclusive answers to some questions about definability in analytic languages that arose shortly after the work by Denef and van den Dries [DD] on p-adic subanalytic sets, and we continue the study of non-archimedean fields with analytic structure of [LR3], [CLR1] and [CL]. We show that the language LK​ consisting of the language of valued fields together with all strictly convergent power series over a complete, rank one valued field K can be expanded, in a definitional way, to a larger language corresponding to an analytic structure (with separated power series) from [CL], hence inheriting all properties from loc. cit., including geometric properties for the definable sets like certain forms of quantifier elimination. Our expansion comes from adding specific, existentially definable functions, which are solutions of certain henselian systems of equations. Moreover, we show that, even when K is algebraically closed, one does not have quantifier elimination in LK​ itself, and hence, passing to expansions is unavoidable in general. We pursue this study in the wider generality of extending non-separated power series rings to separated ones, and give new examples, in particular of the analytic structure over Z[[t]] that can be interpreted and understood now in all complete valued fields. In a separate direction, we show in rather large generality that Weierstrass preparation implies Weierstrass division.

Détail

DOI : 10.4171/jems/662

Classification : 32-XX, 03-XX, 28-XX
Keywords: Henselian valued fields, Tate algebras, strictly convergent power series, subanalytic sets, quantifier elimination, analytic structure, separated power series, non-archimedean geometry, Weierstrass Preparation and Division, Artin approximation, Weierstrass systems