A directed -group that is not a group of divisibility
A hypervaluation of a ring onto a totally ordered non-cancellative semigroup without zero divisors.
A local characterization of Prüfer rings.
A measure theoretic approach to logical quantification
A note on approximation theorems
A two-dimensional domain whose integral closure is not t-linked.
Abhyankar places admit local uniformization in any characteristic
Absolutely S-domains and pseudo-polynomial rings
A domain R is called an absolutely S-domain (for short, AS-domain) if each domain T such that R ⊆ T ⊆ qf(R) is an S-domain. We show that R is an AS-domain if and only if for each valuation overring V of R and each height one prime ideal q of V, the extension R/(q ∩ R) ⊆ V/q is algebraic. A Noetherian domain R is an AS-domain if and only if dim (R) ≤ 1. In Section 2, we study a class of R-subalgebras of R[X] which share many spectral properties with the polynomial ring R[X] and which we call pseudo-polynomial...
Algébricité des fonctions méromorphes prenant certaines valeurs algébriques
An approximation theorem for Dubrovin valuation rings.
Approximate roots of a valuation and the Pierce-Birkhoff conjecture
In this paper, we construct an object, called a system of approximate roots of a valuation, centered in a regular local ring, which describes the fine structure of the valuation (namely, its valuation ideals and the graded algebra). We apply this construction to valuations associated to a point of the real spectrum of a regular local ring . We give two versions of the construction: the first, much simpler, in a special case (roughly speaking, that of rank 1 valuations), the second – in the case...
Binäre Formen und Primideale.
Carrés cartésiens et anneaux de pseudo-valuation
Clôture intégrale des idéaux et équisingularité
This text has two parts. The first one is the essentially unmodified text of our 1973-74 seminar on integral dependence in complex analytic geometry at the Ecole Polytechnique with J-J. Risler’s appendix on the Łojasiewicz exponents in the real-analytic framework. The second part is a short survey of more recent results directly related to the content of the seminar.The first part begins with the definition and elementary properties of the order function associated to an ideal of a reduced analytic...
Constructions de places réelles et géométrie semi-algébrique
Continuous valuations.
Convexity, valuations and Prüfer extensions in real algebra.
Cyclically valued rings and formal power series
Rings of formal power series with exponents in a cyclically ordered group were defined in [2]. Now, there exists a “valuation” on : for every in and in , we let be the first element of the support of which is greater than or equal to . Structures with such a valuation can be called cyclically valued rings. Others examples of cyclically valued rings are obtained by “twisting” the multiplication in . We prove that a cyclically valued ring is a subring of a power series ring with...
De l’application des méthodes valuatives en algèbre différentielle
La théorie des valuations née des travaux des géomètres et arithméticiens du XIX siècle, fit une apparition tardive et encore peu connue au XX siècle en algèbre différentielle. Dans cet article, à travers les contributions de nombreux auteurs, nous présentons une synthèse des divers apports de la théorie des valuations à l’étude des équations différentielles. Nous insistons sur le caractère unificateur de la théorie des valuations en illustrant comment elles permettent de mettre en parallèle des...
Der allgemeine Approximationssatz für Manisbewertungen.