Some counterexamples in -adic analysis
Spherical completeness with infinitesimals.
In the theory of nonarchimedean normed spaces over valued fields other than R or C, the property of spherical completeness is of utmost importance in several contexts, and it appears to play the role conventional completeness does in some topics of classical functional analysis. In this note we give various characterizations of spherical completeness for general ultrametric spaces, related to but different from the notions of pseudo-convergent sequence and pseudo-limit introduced by Ostrowski in...
Strict topologies in non-archimedean function spaces.
Strictly analytic functions on p-adic analytic open sets.
Let K be an algebraically closed complete ultrametric field. M. Krasner and P. Robba defined theories of analytic functions in K, but when K is not spherically complete both theories have the disadvantage of containing functions that may not be expanded in Taylor series in some disks. On other hand, affinoid theories are only defined in a small class of sets (union of affinoid sets) [2], [13] and [17]. Here, we suppose the field K topologically separable (example Cp). Then, we give a new definition...
Subfields of henselian valued fields
Let (K,v) be a henselian valued field of arbitrary rank which is not separably closed. Let k be a subfield of K of finite codimension and be the valuation obtained by restricting v to k. We give some necessary and sufficient conditions for to be henselian. In particular, it is shown that if k is dense in its henselization, then is henselian. We deduce some well known results proved in this direction through other considerations.
Suites de Honda
Sur la constante d’Eisenstein
On cherche à donner une méthode effective de calcul de la constante d’Eisenstein [3] d’une fonction algébrique. On commence en précisant les liens entre cette constante et les rayons de convergence -adiques de la fonction pour les différents nombres premiers . Puis on donne une démonstration entièrement effective du résultat bien connu liant fonctions algébriques et diagonales de fractions rationnelles. Enfin on explique comment en déduire une méthode de calcul générale. On illustre la méthode...
Sur les automorphismes continus d'extensions transcendantes valuées.
Sur les automorphismes d'extensions transcendantes de corps valués
Symmetric and Asymmetric Gaps in Some Fields of Formal Power Series
2000 Mathematics Subject Classification: 03E04, 12J15, 12J25.We consider a construction of fields with symmetric gaps that are not semi-η1. By this construction we give examples of fields with different asymmetric gaps.
Systèmes différentiels linéaires -adiques, structure de Frobenius faible