Journées en l'honneur de Max Dickmann

L'Equipe de Logique Mathématique organise, avec le soutien de l'Institut Mathématique de Jussieu et du réseau européen MODNET, deux journées en l'honneur de Max Dickmann. Ces journées bénéficient également d'une aide financière de l'Université Paris 6.

Album Photo des Journées Dickmann

Programme

Lundi 25 juin

Mardi 26 juin

Organisatrices : Françoise Delon et Danielle Gondard

Titres et résumés

Abstract: What we know and what we would like to know on some well known problems on the semianalytic sets globally defined.

Vincent Astier (Université de Constance) Zéros génériques et groupes spéciaux.

Résumé : Nous présentons quelques résultats allant dans le sens d'une généralisation aux groupes spéciaux de la théorie des zéros génériques des formes quadratiques.

Generic splitting and special groups.

Abstract: We will present some results in the direction of a generalization to special groups of the classical theory of generic splitting of quadratic forms.

Salma Kuhlmann (Université du Saskatchewan) Restricted Exponentiation in Fields of Algebraic Power Series.

Abstract: Ron Brown (1971) proved that a valued vector space of countable dimension admits a valuation basis. This result was applied by S. Kuhlmann (2000) to show that every countable real closed field admits a restricted exponential function, that is, an order preserving isomorphism from the ideal of infinitesimals $(\mathcal{M}_K, +)$ onto the group of 1-units $(1 + \mathcal{M}_K, \cdot)$. A natural question arose whether every real closed field admits a restricted exponential function. In this talk, we give a partial answer to this question. To this end, we investigate valued fields which admit a valuation basis (as valued vector spaces over a given countable ground field $K$). We isolate a property (called TDRP in our paper) for a valued subfield $L$ of a field of generalized power series $F((G))$ (where $G$ is a countable ordered abelian group and $F$ is a real closed, or algebraically closed field). We show that this property implies the existence of a $K$--valuation basis for $L$. In particular, we deduce that the field of rational functions $F(G)$ (the quotient field of the group ring $F[G]$) and the field $\ac{F(G)}$ of power series in $F((G))$ algebraic over $F(G)$ admit $K$--valuation bases. If moreover $F$ is archimedean and $G$ is divisible, we conclude that the real closed field $\ac{F(G)}$ admits a restricted exponential function.

François Lucas (Université d'Angers) Conjectures liées à la conjecture de Pierce-Birkhoff.

Résumé : La conjecture de connexité entraîne la conjecture de Pierce-Birkhoff (toute fonction réelle polynomiale par morceaux est sup-inf de polynômes). Un théorème de connexité et des pistes pour répondre aux conjectures.

Francesco Miraglia (Université de Sao Paulo) Special Groups and Rings of Continuous Functions

Abstract: We shall discuss transversality results for rings of continuous real valued functions and applications to the real holomorphy ring of a formally real field. We shall also show that any reduced special group can be represented as the special group associated to certain subgroup of units of a ring of continuous functions over a Boolean space.

Daniel Plaumann (Université de Constance) Sums of squares on reducible real curves.

Abstract: Given a real algebraic curve C, we ask whether every non-negative polynomial function on C is a sum of squares in the coordinate ring. Scheiderer has classified all irreducible curves for which this is the case. We will discuss how the answer for reducible curves depends on the configuration of the irreducible components.

Alexander Prestel (Université de Constance) Positive Elimination in Valued Fields

Abstract: Let K be an algebraically closed field with a valuation ring O or a real closed field with a convex valuation ring O. We show that the projection of a basic subset of Kn ×Om to Kn is again basic. A basic set is given by a finite union of finite intersections of sets defined by = (or, in the case of a real closed field, the order relation) and divisibility in the valuation ring O.

Marie-Françoise Roy (Université de Rennes I) Décider la positivité : deux approches.

Résumé : Pour un polynôme unitaire en une variable il y a deux manières de décider qu'il est positif : montrer qu'il n'a pas de racine réelle, ou l'écrire sous forme d'une somme de carrés. Ces deux approches sont elles aussi différentes qu'il y paraît ? Fonctionnent-elles encore en plusieurs variables ? Une des deux approches est-elle intrinsèquement plus complexe d'un point de vue algorithmique que l'autre ?

Marcus Tressl (Université de Ratisbonne) Real closed valuation rings

Abstract: A real closed valuation ring is a convex subring of a real closed field. I will give an overview of the algebraic theory and the applications in (real) geometry of these rings, with special emphasis on the Cherlin-Dickmann theorem.

Pour se rendre au colloque