Séminaires : Structures algébriques ordonnées

Equipe(s) : lm,
Responsables :F. Delon, M. Dickmann, D. Gondard
Email des responsables : dickmann@math.univ-paris-diderot.fr
Salle : 1013
Adresse :Sophie Germain

Mardi de 14h00 à 15h45
Page du séminaire et programme
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Titre Ordered transexponential fields
Horaire14:00 à 15:45
Sebastian Krapp (Universität Konstanz, Allemagne)

(Joint work with Salma Kuhlmann)
Studying the growth properties of definable functions in o-minimal settings, Miller established the following remarkable growth dichotomy: an o-minimal expansion of an ordered field is either power bounded or admits a definable exponential function (see [2]). Going one step further in the hierarchy of growth, Miller's dichotomy result naturally led to the question whether there exist o-minimal expansions of ordered fields that are not exponentially bounded. Recent research activity in this area is therefore motivated by the search for either an o-minimal expansion of an ordered exponential field by a transexponential function that eventually exceeds any iterate of the exponential or, contrarily, for a proof that any o-minimal expansion of an ordered field is already exponentially bounded.
In this talk, I will present our axiomatic approach towards the study of ordered fields equipped with a transexponential function from [1]. Namely, denoting by e an exponential and by T a compatible transexponential, we establish a first-order theory of ordered transexponential fields in which the functional equation $T(x + 1) = e(T(x))$ holds for any positive $x$. While the archimedean models of this theory are readily described, the study of the non-archimedean models leads to a systematic examination of the induced structure on the residue field and the value group under the natural (i.e. the finest non-trivial convex) valuation. Moreover, I will illustrate construction methods for transexponentials on non-archimedean ordered exponential fields.

All relevant valuation theoretic background will be introduced.

[1] L. S. Krapp and S. Kuhlmann, ˜Ordered transexponential fields™, preprint, 2023, arXiv:2305.04607v2.
[2] C. Miller, "A Growth Dichotomy for O-minimal Expansions of Ordered Fields™, Logic: from Foundations to Applications (eds W. Hodges, M. Hyland, C. Steinhorn and J. Truss; Oxford Sci. Publ., Oxford Univ. Press, New York, 1996) 385–399.
AdresseSophie Germain