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Franz-Viktor Kuhlmann

Publications and source records attributed to Franz-Viktor Kuhlmann.

At least 55 records · Page 3Linked to original sources
arXiv2013

Towers of complements to valuation rings and truncation closed embeddings of valued fields

We study necessary and sufficient conditions for a valued field $\KF$ with value group $G$ and residue field $\kf$ (with char $\KF$ = char $\kf$) to admit a truncation closed embedding in the field of generalized power series $\kf((G, f))$ (with factor set $f$). We show that this is equivalent to the existence of a family ({\it tower of complements}) of $\kf$-subspaces of $\KF$ which are complements of the (possibly fractional) ideals of the valuation ring. If $\KF$ 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.

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arXiv2013

Algebraic independence of elements in immediate extensions of valued fields

Refining a constructive combinatorial method due to MacLane and Schilling, we give several criteria for a valued field that guarantee that all of its maximal immediate extensions have infinite transcendence degree. If the value group of the field has countable cofinality, then these criteria give the same information for the completions of the field. The criteria have applications to the classification of valuations on rational function fields. We also apply the criteria to the question which extensions of a maximal valued field, algebraic or of finite transcendence degree, are again maximal. In the case of valued fields of infinite $p$-degree, we obtain the worst possible examples of nonuniqueness of maximal immediate extensions: fields which admit an algebraic maximal immediate extension as well as one of infinite transcendence degree.

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arXiv2013

A common generalization of metric, ultrametric and topological fixed point theorems - alternative version

We present a general fixed point theorem which can be seen as the quintessence of the principles of proof for Banach's Fixed Point Theorem, ultrametric and certain topological fixed point theorems. It works in a minimal setting, not involving any metrics. We demonstrate its applications to the metric, ultrametric and topological cases, and to ordered abelian groups and fields.

math.AC↗
arXiv2013

Defects of Algebraic Function Fields, Completion Defects and Defect Quotients

The {\it defect} (also called {\it ramification deficiency}) of valued field extensions is a major stumbling block in deep open problems of valuation theory in positive characteristic. For a detailed analysis, we define and investigate two weaker notions of defect: the {\it completion defect} and the {\it defect quotient}. We define them for finite extensions as well as for certain valued function fields (those with "Abhyankar valuations" that are nontrivial on the ground field). We investigate both defects and present analogues of the results that hold for the usual defect.

math.AC↗
arXiv2013

The relative approximation degree in valued function fields

We continue the work of Kaplansky on immediate valued field extensions and determine special properties of elements in such extensions. In particular, we are interested in the question when an immediate valued function field of transcendence degree 1 is henselian rational (i.e., generated, modulo henselization, by one element). If so, then wild ramification can be eliminated in this valued function field. The results presented in this paper are crucial for the first author's proof of henselian rationality over tame fields, which in turn is used in his work on local uniformization.

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arXiv2013

Embedding theorems for spaces of $\R$-places of rational function fields and their products

We study spaces $M(R(y))$ of $\R$-places of rational function fields $R(y)$ in one variable. For extensions $F|R$ of formally real fields, with $R$ real closed and satisfying a natural condition, we find embeddings of $M(R(y))$ in $M(F(y))$ and prove uniqueness results. Further, we study embeddings of products of spaces of the form $M(F(y))$ in spaces of $\R$-places of rational function fields in several variables. Our results uncover rather unexpected obstacles to a positive solution of the open question whether the torus can be realized as a space of $\R$-places.

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arXiv2010

Characterization of Extremal Valued 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.

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arXiv2010

The defect

We give an introduction to the valuation theoretical phenomenon of "defect", also known as "ramification deficiency". We describe the role it plays in deep open problems in positive characteristic: local uniformization (the local form of resolution of singularities), the model theory of valued fields, the structure theory of valued function fields. We give several examples of algebraic extensions with non-trivial defect. We indicate why Artin-Schreier defect extensions play a central role and describe a way to classify them. Further, we give an overview of various results about the defect that help to tame or avoid it, in particular "stability" theorems and theorems on "henselian rationality", and show how they are applied. Finally, we include a list of open problems.

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arXiv2010

A classification of Artin-Schreier defect extensions and a characterization of defectless fields

We classify Artin-Schreier extensions of valued fields with non-trivial defect according to whether they are connected with purely inseparable extensions with non-trivial defect, or not. We use this classification to show that in positive characteristic, a valued field is algebraically complete if and only if it has no proper immediate algebraic extension and every finite purely inseparable extension is defectless. This result is an important tool for the construction of algebraically complete fields. We also consider extremal fields (= fields for which the values of the elements in the images of arbitrary polynomials always assume a maximum). We characterize inseparably defectless, algebraically maximal and separable-algebraically maximal fields in terms of extremality, restricted to certain classes of polynomials. We give a second characterization of algebraically complete fields, in terms of their completion. Finally, a variety of examples for Artin-Schreier extensions of valued fields with non-trivial defect is presented.

math.AC↗
arXiv2010

Approximation of elements in henselizations

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 $K$.

math.AC↗
arXiv2010

Maps on ultrametric spaces, Hensel's Lemma, and differential equations over valued fields

We give a criterion for maps on ultrametric spaces to be surjective and to preserve spherical completeness. We show how Hensel's Lemma and the multi-dimensional Hensel's Lemma follow from our result. We give an easy proof that the latter holds in every henselian field. We also prove a basic infinite-dimensional Implicit Function Theorem. Further, we apply the criterion to deduce various versions of Hensel's Lemma for polynomials in several additive operators, and to give a criterion for the existence of integration and solutions of certain differential equations on spherically complete valued differential fields, for both valued D-fields in the sense of Scanlon, and differentially valued fields in the sense of Rosenlicht. We modify the approach so that it also covers logarithmic-exponential power series fields. Finally, we give a criterion for a sum of spherically complete subgroups of a valued abelian group to be spherically complete. This in turn can be used to determine elementary properties of power series fields in positive characteristic.

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arXiv2010

Elimination of Ramification I: The Generalized Stability Theorem

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.

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arXiv2010

Dense subfields of henselian fields, and integer parts

We show that every henselian valued field $L$ of residue characteristic 0 admits a proper subfield $K$ which is dense in $L$. We present conditions under which this can be taken such that $L|K$ is transcendental and $K$ is henselian. These results are of interest for the investigation of integer parts of ordered fields. We present examples of real closed fields which are larger than the quotient fields of all their integer parts. Finally, we give rather simple examples of ordered fields that do not admit any integer part and of valued fields that do not admit any subring which is an additive complement of the valuation ring.

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arXiv2010

Additive Polynomials and Their Role in the Model Theory of Valued Fields

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 in $K$. Finally, we prove that every minimal purely wild extension of a henselian valued field is generated by a $p$-polynomial.

math.AC↗
arXiv2010

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.

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arXiv2010

Places of algebraic function fields in arbitrary characteristic

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.

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