Luc Bélair

Model theory of the Frobenius on the Witt vectors

We give axiomatizations and prove quantifier elimination theorems for first-order theories of unramified valued fields with an automorphism having a close interaction with the valuation. We achieve an analogue of the classical Ostrowski theory of pseudoconvergence. In the outstanding case of Witt vectors with their Frobenius map, we use the ∂-ring formalism from Joyal.

Rings of p -adic functions

We study first-order properties of the quotient rings (V)/ by a prime ideal where (V) is the ring of p-adic valued continuous definable functions on some affine p-adic variety V. We show that they are integrally closed Henselian local rings, with a p-adically closed residue field and field of fractions, and they are not valuation rings in general but always satisfy ∀ x, y(x∣y 2 ∨ y∣x 2).

La définissabilité des entiers dans les corps de courbes réelles archimédiens

Resume On montre que les entiers naturels sont definissables dans les corps de courbes reelles sur un corps ordonnable qui admet au moins un ordre archimedien. Ceci generalise le resultat de Raphael Robinson sur les corps de fonctions rationnelles. Pour citer cet article : L. Belair, C. R. Acad. Sci. Paris, Ser. I 336 (2003). Abstract We show that the natural numbers are definable in the function field of a curve over a formally real field which admits at least one Archimedean order. This generalises Raphael Robinsons result on fields of rational functions. To cite this article: L. Belair, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Separably closed fields and contractive ore modules

We consider valued fields with a distinguished contractive map as valued modules over the Ore ring of difference operators. We prove quantifier elimination for separably closed valued fields with the Frobenius map, in the pure module language augmented with functions yielding components for a p-basis and a chain of subgroups indexed by the valuation group.

Pseudovaluation domains with Vapnik-Chervonenkis classes of definable sets

We use model-theoretic methods to give examples of pseudovaluation domains with Vapnik-Chervonenkis classes of definable sets.

Infinitesimally stable theories of henselian rings

Abstract We consider coherent theories of henselian rings which are infinitesimally stable, i.e. stable for the etale factorisation system. We give continuum many examples as stabilisations of model-complete complete theories of fields.

Elementary equivalence and codimension in p-adic fields

We give examples of fields elementarily equivalent to a given finite extension of the p-adic numbers but not containing a subfield of finite codimension elementarily equivalent to the p-adics.