Towards a Model Theory for Transseries

The differential field of transseries extends the field of real Laurent series, and occurs in various context: asymptotic expansions, analytic vector fields, o-minimal structures, to name a few. We give an overview of the algebraic and model-theoretic aspects of this differential field, and report on our efforts to understand its first-order theory.Comment: Notre Dame J. Form. Log., to appear; 33 p

Henselian Local Rings: Around a Work in Progress

I shall outline an elementary and effective construction of the Henselization of a local ring (which could be implemented in some computer algebra systems) and an effective proof of several classical results about Henselian local rings

The Abhyankar-Jung Theorem

We show that every quasi-ordinary Weierstrass polynomial P(Z) = Z^d+a_1 (X) Z^{d-1}+...+a_d(X) \in \K[[X]][Z] , $X=(X_1,..., X_n)$, over an algebraically closed field of characterisic zero \K, and satisfying $a_1=0$, is $\nu$-quasi-ordinary. That means that if the discriminant \Delta_P \in \K[[X]] is equal to a monomial times a unit then the ideal $(a_i^{d!/i}(X))_{i=2,...,d}$ is principal and generated by a monomial. We use this result to give a constructive proof of the Abhyankar-Jung Theorem that works for any Henselian local subring of \K[[X]] and the function germs of quasi-analytic families.Comment: 14 pages. The toric case has been added. To be published in Journal of Algebr

Characterizing Diophantine Henselian valuation rings and valuation ideals

We give a characterization, in terms of the residue field, of those henselian valuation rings and those henselian valuation ideals that are diophantine. This characterization gives a common generalization of all the positive and negative results on diophantine henselian valuation rings and diophantine valuation ideals in the literature. We also treat questions of uniformity and we apply the results to show that a given field can carry at most one diophantine nontrivial equicharacteristic henselian valuation ring or valuation ideal

Local Bézout Theorem

AbstractWe give an elementary proof of what we call the Local Bézout Theorem. Given a system of n polynomials in n indeterminates with coefficients in a Henselian local domain, (V,m,k), which residually defines an isolated point in kn of multiplicity r, we prove (under some additional hypothesis on V) that there are finitely many zeroes of the system above the residual zero (i.e., with coordinates in m), and the sum of their multiplicities is r. Our proof is based on techniques of computational algebra

Panorama of p-adic model theory

ABSTRACT. We survey the literature in the model theory of p-adic numbers since\ud Denef’s work on the rationality of Poincaré series. / RÉSUMÉ. Nous donnons un aperçu des développements de la théorie des modèles\ud des nombres p-adiques depuis les travaux de Denef sur la rationalité de séries de Poincaré,\ud par une revue de la bibliographie

05021 Abstracts Collection -- Mathematics, Algorithms, Proofs

From 09.01.05 to 14.01.05, the Dagstuhl Seminar 05021 ``Mathematics, Algorithms, Proofs\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. LinkstFo extended abstracts or full papers are provided, if available