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We prove in arbitrary characteristic that an immediate valued algebraic function field $F$ of transcendence degree 1 over a tame field $K$ is contained in the henselization of $K(x)$ for a suitably chosen $x\in F$. This eliminates ramification in such valued function fields. We give generalizations of this result, relaxing the assumption on $K$. Our theorems have important applications to local uniformization and to the model theory of valued fields in positive and mixed characteristic., new improved version

We study and characterize the class of valuations on rational functions fields that are invariant under permutation of the variables and can be extended to valuations with the same property whenever a finite number of new variables is adjoined. The Gaus valuation is in this class, which constitutes a natural generalization of the concept of Gaus valuation. Further, we apply our characterization to show that the most common ad hoc generalization of the Gaus valuation is also in this class.

A basic version of Abhyankar's Lemma states that for two finite extensions $L$ and $F$ of a local field $K$, if $L|K$ is tamely ramified and if the ramification index of $L|K$ divides the ramification index of $F|K$, then the compositum $L.F$ is an unramified extension of $F$. In this paper, we generalize the result to valued fields with value groups of rational rank 1, and show that the latter condition is necessary. Replacing the condition on the ramification indices by the condition that the value group of $L$ be contained in that of $F$, we generalize the result further in order to give a necessary and sufficient condition for the elimination of tame ramification of an arbitrary extension $F|K$ by a suitable algebraic extension of the base field $K$. In addition, we derive more precise ramification theoretical statements and give several examples.

It is proved in this article that the theory of separably closed nontrivially valued fields of characteristic p > 0 and imperfection degree e > 0 (e ≤ ∞) has quantifier elimination in the language ${{\cal L}_{p,{\rm{div}}}} = \{ + , - , \times ,0,1\} \cup {\{ {\lambda _{n,j}}(x;{y_1}, \ldots ,{y_n})\} _{0 \le n < \omega ,0 \le j < {p^n}}} \cup \{ |\}$; in particular, when e is finite, the corresponding theory has quantifier elimination in the language ${\cal L} = \{ + , - , \times ,0,1\} \cup \{ {b_1}, \ldots ,{b_e}\} \cup {\{ {\lambda _{e,j}}(x;{b_1}, \ldots ,{b_e})\} _{0 \le j < {p^e}}} \cup \{ |\}$.

We extend the characterization of extremal valued fields given in [2] to the missing case of valued fields of mixed characteristic with perfect residue field. This leads to a complete characterization of the tame valued fields that are extremal. The key to the proof is a model theoretic result about tame valued fields in mixed characteristic. Further, we prove that in an extremal valued field of finitep-degree, the images of all additive polynomials have the optimal approximation property. This fact can be used to improve the axiom system that is suggested in [8] for the elementary theory of Laurent series fields over finite fields. Finally we give examples that demonstrate the problems we are facing when we try to characterize the extremal valued fields with imperfect residue fields. To this end, we describe several ways of constructing extremal valued fields; in particular, we show that in every ℵ1saturated valued field the valuation is a composition of extremal valuations of rank 1.

A henselian valued field $K$ is called separably tame if its separable-algebraic closure $K^{\operatorname{sep}}$ is a tame extension, that is, the ramification field of the normal extension $K^{\operatorname{sep}}|K$ is separable-algebraically closed. Every separable-algebraically maximal Kaplansky field is a separably tame field, but not conversely. In this paper, we prove Ax-Kochen-Ershov Principles for separably tame fields. This leads to model completeness and completeness results relative to the value group and residue field. As the maximal immediate extensions of separably tame fields are in general not unique, the proofs have to use much deeper valuation theoretical results than those for other classes of valued fields which have already been shown to satisfy Ax-Kochen-Ershov Principles. Our approach also yields alternate proofs of known results for separably closed valued fields., 30 pages. arXiv admin note: substantial text overlap with arXiv:1304.0194

We take a fresh look at the important Caristi-Kirk Fixed Point Theorem and link it to the recently developed theory of ball spaces, which provides generic fixed point theorems for contracting functions in a number of applications including, but not limited to, metric spaces. The connection becomes clear from a proof of the Caristi-Kirk Theorem given by J.-P. Penot in 1976. We define Caristi-Kirk ball spaces and use a generic fixed point theorem to reprove the Caristi-Kirk Theorem. Further, we show that a metric space is complete if and only if all of its Caristi-Kirk ball spaces are spherically complete.

We present a general structure theorem for the Hardy field of an o-minimal expansion of the reals by restricted analytic functions and an unrestricted exponential. We proceed to analyze its residue fields with respect to arbitrary convex valuations, and deduce a power series expansion of exponential germs. We apply these results to cast "Hardy's conjecture" (see \cite[p.111]{[KS]}) in a more general framework. This paper is a follow up to \cite{[K-K2]} and is partially based on unpublished results of \cite{[K-K]}. A previous version \cite{[K-K1]} (which was dedicated to Murray A. Marshall on his 60th birthday) remained unpublished. In \cite{[W]} our structure theorem for the residue fields was rediscovered and applied to the diophantine context. Due to this revived interest, we decided to rework the preprint \cite{[K-K1]} and submit it to the Proceedings Volume.

A henselian valued field K is called a tame field if its algebraic closure K ~ ${\tilde{K}}$ is a tame extension, that is, the ramification field of the normal extension K ~ | K ${\tilde{K}|K}$ is algebraically closed. Every algebraically maximal Kaplansky field is a tame field, but not conversely. We develop the algebraic theory of tame fields and then prove Ax–Kochen–Ershov Principles for tame fields. This leads to model completeness and completeness results relative to value group and residue field. As the maximal immediate extensions of tame fields will in general not be unique, the proofs have to use much deeper valuation theoretical results than those for other classes of valued fields which have already been shown to satisfy Ax–Kochen–Ershov Principles. The results of this paper have been applied to gain insight in the Zariski space of places of an algebraic function field, and in the model theory of large fields.

We characterize those valued fields for which the image of the valuation ring under every polynomial in several variables contains an element of maximal value, or zero., Comment: 12 pages

We study necessary and sufficient conditions for a valued field K with value group G and residue field k (with char K = char k ) to admit a truncation closed embedding in the field of generalized power series k ( ( G , f ) ) (with factor set f). We show that this is equivalent to the existence of a family (tower of complements) of k-subspaces of K which are complements of the (possibly fractional) ideals of the valuation ring, and satisfying certain natural conditions. If K is a Henselian field of characteristic 0 or, more generally, an algebraically maximal Kaplansky field, we give an intrinsic construction of such a family which does not rely on a given truncation closed embedding. We also show that towers of complements and truncation closed embeddings can be extended from an arbitrary field to at least one of its maximal immediate extensions.

We generalize the main result of [K–K–S] concerning the convex embeddings of a chain Γ in a lexicographic power Δ Γ . For a fixed non-empty chain Δ, we derive necessary and sufficient conditions for the existence of non-empty solutions Γ to each of the lexicographic functorial equations (Δ Γ ) ≤0 � Γ , (Δ Γ ) � Γa nd (Δ Γ )

We prove a general version of the "Stability Theorem": if $K$ is a valued field such that the ramification theoretical defect is trivial for all of its finite extensions, and if $F|K$ is a finitely generated (transcendental) extension of valued fields for which equality holds in the Abhyankar inequality, then the defect is also trivial for all finite extensions of $F$. This theorem is applied to eliminate ramification in such valued function fields. It has applications to local uniformization and to the model theory of valued fields in positive characteristic., Comment: 31 pages

We consider the Zariski space of all places of an algebraic function field $F|K$ of arbitrary characteristic and investigate its structure by means of its patch topology. We show that certain sets of places with nice properties (e.g., prime divisors, places of maximal rank, zero-dimensional discrete places) lie dense in this topology. Further, we give several equivalent characterizations of fields that are large, in the sense of F. Pop's Annals paper {\it Embedding problems over large fields}. We also study the question whether a field $K$ is existentially closed in an extension field $L$ if $L$ admits a $K$-rational place. In the appendix, we prove the fact that the Zariski space with the Zariski topology is quasi-compact and that it is a spectral space., Comment: 27 pages

For valued fields $K$ of rank higher than 1, we describe how elements in the henselization $K^h$ of $K$ can be approximated from within $K$; our result is a handy generalization of the well-known fact that in rank 1, all of these elements lie in the completion of $K$. We apply the result to show that if an element $z$ algebraic over $K$ can be approximated from within $K$ in the same way as an element in $K^h$, then $K(z)$ is not linearly disjoint from $K^h$ over $K$., Comment: 13 pages

We discuss the role of additive polynomials and p-polynomials in the theory of valued fields of positive characteristic and in their model theory. We outline the basic properties of rings of additive polynomials and discuss properties of valued fields of positive characteristic as modules over such rings. We prove the existence of Frobenius-closed bases of algebraic function fields F|K in one variable and deduce that F/K is a free module over the ring of additive polynomials with coefficients inK. Finally, we prove that every minimal purely wild extension of a henselian valued field is generated by a p-polynomial.

We prove local uniformization of Abhyankar valuations of an algebraic function field K over a ground field k. Our result generalizes the proof of this result, with the additional assumption that the residue field of the valuation ring is separable over k, by Hagen Knaf and Franz-Viktor Kuhlmann. The proof in this paper uses different methods, being inspired by the approach of Zariski and Abhyankar., Comment: 28 pages. Final Final version

International audience; We survey some important properties of fields of generalized series and of exponential-logarithmic series, with particular emphasis on their possible differential structure, based on a joint work of the author with S. Kuhlmann [KM12b,KM11].