Franz-Viktor Kuhlmann

Value groups, residue fields, and bad places of rational function fields

We classify all possible extensions of a valuation from a ground field K to a rational function field in one or several variables over K. We determine which value groups and residue fields can appear, and we show how to construct extensions having these value groups and residue fields. In particular, we give several constructions of extensions whose corresponding value group and residue field extensions are not finitely generated. In the case of a rational function field K(x) in one variable, we consider the relative algebraic closure of K in the henselization of K(x) with respect to the given extension, and we show that this can be any countably generated separable-algebraic extension of K. In the tame case, we show how to determine this relative algebraic closure. Finally, we apply our methods to power series fields and the p-adics.

Valuation Theoretic and Model Theoretic Aspects of Local Uniformization

In this paper, I will take you on an excursion from Algebraic Geometry through Valuation Theory to Model Theoretic Algebra, and back. If our destination sounds too exotic for you, you may jump off at the Old World (Valuation Theory) and divert yourself with problems and examples until you catch our plane back home.

Immediate and purely wild extensions of valued fields

Kaplanskys hypothesis A concerning valued fields is put into a Galois theoretic setting. Accordingly, Kaplanskys theorem on maximal immediate extensions can be deduced from the Schur-Zassenhaus theorem about conjugacy of complements in profinite groups. Some generalization of Kaplanskys theory is given, concerning maximal purely wild extensions.

Exponentiation in power series fields

We prove that for no nontrivial ordered abelian group G, the ordered power series fleld R((G)) admits an exponential, i.e. an isomorphism between its ordered additive group and its ordered multiplicative group of positive elements, but that there is a non- surjective logarithm. For an arbitrary ordered fleld k, no exponen- tial on k((G)) is compatible, that is, induces an exponential on k through the residue map. This is proved by showing that certain functional equations for lexicographic powers of ordered sets are not solvable.

Elementary properties of power series fields over finite fields

In spite of the analogies between Qp and FFp ((t)) which became evident through the work of Ax and Kochen, an adaptation of the complete recursive axiom system given by them for Q(p to the case of Fp ((t)) does not render a complete axiom system. We show the independence of elementary properties which express the action of additive polynomials as maps on FFp ((t)). We formulate an elementary property expressing this action and show that it holds for all maximal valued fields. We also derive an example of a rather simple immediate valued function field over a henselian defectless ground field which is not a henselian rational function field. This example is of special interest in connection with the open problem of local uniformization in positive characteristic.

Quantifier elimination for henselian fields relative to additive and multiplicative congruences

Several classes of henselian valued fields admit quantifier elimination relative to structures which reflect the additive and multiplicative congruences of the field. Value groups and residue fields may be viewed as reduts of these structures. A general theorem is given using the theory of tame extensions of henselian fields. Special cases like the case ofp-adically closed fields and the case of henselian fields of residue characteristic 0 are discussed.

An isomorphism theorem for henselian algebraic extensions of valued fields

In general, the value groups and the residue fields do not suffice to classify the algebraic henselian extensions of a valued fieldK, up to isomorphism overK. We define a stronger, yet natural structure which carries information about additive and multiplicative congruences in the valued field, extending the information carried by value groups and residue fields. We discuss the cases where these “mixed structures” give a solution of the classification problem.

Valuation theory and its applications

A note on tame fields by K. Aghigh and S. K. Khanduja Some remarks about asymptotic couples by M. Aschenbrenner Prime segments for cones and rings by H. H. Brungs, H. Marubayashi, and E. Osmanagic Irreducibility criterion: A geometric point of view by V. Cossart and G. Moreno-Socias On the decidability of the existential theory of

Embedding ordered fields in formal power series fields

{\mathbb F_p}[[t]]

Maps on Ultrametric Spaces, Hensel's Lemma, and Differential Equations Over Valued Fields

by J. Denef and H. Schoutens Galois groups over nonrigid fields by W. Gao, D. B. Leep, J. Minac, and T. L. Smith Automorphisms of formal power series rings over a valuation ring by B. Green Regular curves over Prufer domains by H. Knaf Encoding valuations in absolute Galois groups by J. Koenigsmann Dynamic computations inside the algebraic closure of a valued field by F.-V. Kuhlmann, H. Lombardi, and H. Perdry Preorders, rings, lattice-ordered groups and formal power series by G. Leloup The theorem of Grunwald-Wang in the setting of valuation theory by F. Lorenz and P. Roquette Invariants of singular plane curves by R. I. Michler