arXiv subjects

Salma Kuhlmann

Publications and source records attributed to Salma Kuhlmann.

At least 37 records · Page 2Linked to original sources

arXiv2022

An intrinsic characterization of moment functionals in the compact case

We consider the class of all linear functionals $L$ on a unital commutative real algebra $A$ that can be represented as an integral w.r.t. to a Radon measure with compact support in the character space of $A$. Exploiting a recent generalization of the classical Nussbaum theorem, we establish a new characterization of this class of moment functionals solely in terms of a growth condition intrinsic to the given linear functional. To the best of our knowledge, our result is the first to exactly identify the compact support of the representing Radon measure. We also describe the compact support in terms of the largest Archimedean quadratic module on which $L$ is non-negative and in terms of the smallest submultiplicative seminorm w.r.t. which $L$ is continuous. Moreover, we derive a formula for computing the measure of each singleton in the compact support, which in turn gives a necessary and sufficient condition for the support to be a finite set. Finally, some aspects related to our growth condition for topological algebras are also investigated.

math.FA↗

arXiv2022

The automorphism group of a valued field of generalised formal power series

Let $ k $ be a field, $ G $ a totally ordered abelian group and $ \mathbb K = k((G)) $ the maximal field of generalised power series, endowed with the canonical valuation $ v $. We study the group $ v \mathrm{-Aut} K $ of valuation preserving automorphisms of a subfield $ k(G)\subseteq K\subseteq \mathbb K $, where $ k(G) $ is the fraction field of the group ring $ k[G] $. Under the assumption that $ K $ satisfies two lifting properties we are able to generalise and refine Hofberger's decomposition of $ v \mathrm{-Aut}\mathbb K $ and prove a structure theorem decomposing $ v\mathrm{-Aut} K $ into a 4-factor semi-direct product of notable subgroups. We then identify a large class of Hahn fields satisfying the two aforementioned lifting properties. Next we focus on the group of strongly additive automorphisms of $ K $. We give an explicit description of the group of strongly additive internal automorphisms in terms of the groups of homomorphisms $ \mathrm{Hom}(G,k^\times) $ of $ G $ into $ k^\times $ and $ \mathrm{Hom}(G,1+I_K) $ of $ G $ into the group of 1-units of the valuation ring of $ K $. Finally, we specialise our results to some relevant special cases. In particular, we extend the work of Schilling on the field of Laurent series and that of Deschamps on the field of Puiseux series.

math.AC↗

arXiv2022

Projective limits techniques for the infinite dimensional moment problem

We deal with the following general version of the classical moment problem: when can a linear functional on a unital commutative real algebra $A$ be represented as an integral with respect to a Radon measure on the character space $X(A)$ of $A$ equipped with the Borel $σ-$algebra generated by the weak topology? We approach this problem by constructing $X(A)$ as a projective limit of the character spaces of all finitely generated unital subalgebras of $A$. Using some fundamental results for measures on projective limits of measurable spaces, we determine a criterion for the existence of an integral representation of a linear functional on $A$ with respect to a measure on the cylinder $σ-$algebra on $X(A)$ (resp. a Radon measure on the Borel $σ-$algebra on $X(A)$) provided that for any finitely generated unital subalgebra of $A$ the corresponding moment problem is solvable. We also investigate how to localize the support of representing measures for linear functionals on $A$. These results allow us to establish infinite dimensional analogues of the classical Riesz-Haviland and Nussbaum theorems as well as a representation theorem for linear functionals non-negative on a "partially Archimedean" quadratic module of $A$. Our results in particular apply to the case when $A$ is the algebra of polynomials in infinitely many variables or the symmetric tensor algebra of a real infinite dimensional vector space, providing a unified setting which enables comparisons between some recent results for these instances of the moment problem.

math.FA↗

arXiv2021

On Rayner structures

In this note, we study substructures of generalised power series fields induced by families of well-ordered subsets of the group of exponents. We characterise the set-theoretic and algebraic properties of the induced substructures in terms of conditions on the families. We extend the work of Rayner by giving both \emph{necessary} and sufficient conditions to obtain truncation closed subgroups, subrings and subfields.

math.AC↗

arXiv2021

Strongly NIP almost real closed fields

The following conjecture is due to Shelah-Hasson: Any infinite strongly NIP field is either real closed, algebraically closed, or admits a non-trivial definable henselian valuation, in the language of rings. We specialise this conjecture to ordered fields in the language of ordered rings, which leads towards a systematic study of the class of strongly NIP almost real closed fields. As a result, we obtain a complete characterisation of this class.

math.LO↗

arXiv2020

Ordered fields dense in their real closure and definable convex valuations

In this paper, we undertake a systematic model and valuation theoretic study of the class of ordered fields which are dense in their real closure. We apply this study to determine definable henselian valuations on ordered fields, in the language of ordered rings. In light of our results, we re-examine the Shelah-Hasson Conjecture (specialised to ordered fields) and provide an example limiting its valuation theoretic conclusions.

math.LO↗

arXiv2019

On Strongly NIP Ordered Fields and Definable Convex Valuations

We investigate what henselian valuations on ordered fields are definable in the language of ordered rings. This leads towards a systematic study of the class of ordered fields which are dense in their real closure. Some results have connections to recent conjectures on definability of henselian valuations in strongly NIP fields. Moreover, we obtain a complete characterisation of strongly NIP almost real closed fields.

math.LO↗

arXiv2019

Exponential fields and Conway's omega-map

Inspired by Conway's surreal numbers, we study real closed fields whose value group is isomorphic to the additive reduct of the field. We call such fields omega-fields and we prove that any omega-field of bounded Hahn series with real coefficients admits an exponential function making it into a model of the theory of the real exponential field. We also consider relative versions with more general coefficient fields.

math.LO↗

arXiv2018

Definable Valuations induced by multiplicative subgroups and NIP Fields

We study the algebraic implications of the non-independence property (NIP) and variants thereof (dp-minimality) on infinite fields, motivated by the conjecture that all such fields which are neither real closed nor separably closed admit a definable henselian valuation. Our results mainly focus on Hahn fields and build up on Will Johnson's preprint "dp-minimal fields", arXiv: 1507.02745v1, July 2015.

math.LO↗

arXiv2018

On the determinacy of the moment problem for symmetric algebras of a locally convex space

This note aims to show a uniqueness property for the solution (whenever exists) to the moment problem for the symmetric algebra $S(V)$ of a locally convex space $(V, τ)$. Let $μ$ be a measure representing a linear functional $L: S(V)\to\mathbb{R}$. We deduce a sufficient determinacy condition on $L$ provided that the support of $μ$ is contained in the union of the topological duals of $V$ w.r.t. to countably many of the seminorms in the family inducing $τ$. We compare this result with some already known in literature for such a general form of the moment problem and further discuss how some prior knowledge on the support of the representing measure influences its determinacy.

math.FA↗

arXiv2018

Compatibility of quasi-orderings and valuations; A Baer-Krull Theorem for quasi-ordered Rings

In his work of 1969, Merle E. Manis introduced valuations on commutative rings. Recently, the class of totally quasi-ordered rings was developped by the second author. In the present paper, we establish the notion of compatibility between valuations and quasi-orders on rings, leading to a definition of the rank of a quasi-ordered ring. Moreover, we prove a Baer-Krull Theorem for quasi-ordered rings: fixing a Manis valuation v on R, we characterize all v-compatible quasi-orders of R by lifting the quasi-orders from the residue class ring to R itself.

math.AC↗

arXiv2018

The differential rank of a differential-valued field

We develop a notion of (principal) differential rank for differential-valued fields, in analog of the exponential rank and of the difference rank. We give several characterizations of this rank. We then give a method to define a derivation on a field of generalized power series and use this method to show that any totally ordered set can be realized as the principal differential rank of a H-field.

math.AC↗

arXiv2017

Infinite dimensional moment problem: open questions and applications

Infinite dimensional moment problems have a long history in diverse applied areas dealing with the analysis of complex systems but progress is hindered by the lack of a general understanding of the mathematical structure behind them. Therefore, such problems have recently got great attention in real algebraic geometry also because of their deep connection to the finite dimensional case. In particular, our most recent collaboration with Murray Marshall and Mehdi Ghasemi about the infinite dimensional moment problem on symmetric algebras of locally convex spaces revealed intriguing questions and relations between real algebraic geometry, functional and harmonic analysis. Motivated by this promising interaction, the principal goal of this paper is to identify the main current challenges in the theory of the infinite dimensional moment problem and to highlight their impact in applied areas. The last advances achieved in this emerging field and briefly reviewed throughout this paper led us to several open questions which we outline here.

math.FA↗

arXiv2017

A Baer-Krull theorem for quasi-ordered groups

We give group analogs of two important theorems of real algebra concerning convex valuations, one of which is the Baer-Krull theorem. We do this by using quasi-orders, which gives a uniform approach to valued and ordered groups. We also recover the classical Baer-Krull theorem from its group analog.

math.AC↗

arXiv2016

Murray A. Marshall [24.3.1940 - 1.05.2015]: A brief mathematical obituary

On April 22 2015, Murray Marshall was invited by Igor Klep and Victor Vinnikov to talk in the special session on "Operator Theory, Real Algebraic Geometry, and Moment Problems" that they co-organised for IWOTA 2015 in Tbilisi, Georgia, July 6--10. This invitation to an IWOTA edition was considered by the organisers as long overdue, given Marshall's important contributions to Real Algebraic Geometry in general, and to the multi-dimensional Moment Problem in particular. Marshall declined on April 27th writing: "Dear Victor, Thanks very much for the invitation. I am sorry but I must decline. The reason is that I don't enjoy traveling as much as I used to. North America or Europe is about the maximum distance I am willing to dealwith. I hope you are well. Murray." This response startled us, as Marshall has always been a grand traveler when it came to mathematics. Our fears were alas confirmed; Marshall passed away just a few days later. In July, I held a talk at that session myself on joint work with Marshall, and was asked by the editors of the proceedings volume to write this mathematical obituary.

math.HO↗

arXiv2016

The analogue of Hilbert's 1888 theorem for Even Symmetric Forms

Hilbert proved in 1888 that a positive semidefinite (psd) real form is a sum of squares (sos) of real forms if and only if $n=2$ or $d=1$ or $(n,2d)=(3,4)$, where $n$ is the number of variables and $2d$ the degree of the form. We study the analogue for even symmetric forms. We establish that an even symmetric $n$-ary $2d$-ic psd form is sos if and only if $n=2$ or $d=1$ or $(n,2d)=(n,4)_{n \geq 3}$ or $(n,2d)= (3,8)$.

math.AG↗

arXiv2016

Valuation theory of exponential Hardy fields II: Principal parts of germs in the Hardy field of o-minimal exponential expansions of the reals

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.

math.LO↗

arXiv2016

The valuation difference rank of a quasi-ordered difference field

There are several equivalent characterizations of the valuation rank of an ordered or valued field. In this paper, we extend the theory to the case of an ordered or valued {\it difference} field (that is, ordered or valued field endowed with a compatible field automorphism). We introduce the notion of {\it difference rank}. To treat simultaneously the cases of ordered and valued fields, we consider quasi-ordered fields. We characterize the difference rank as the quotient modulo the equivalence relation naturally induced by the automorphism (which encodes its growth rate). In analogy to the theory of convex valuations, we prove that any linearly ordered set can be realized as the difference rank of a maximally valued quasi-ordered difference field. As an application, we show that for every regular uncountable cardinal $κ$ such that $κ= κ^{< κ}$, there are $2^κ$ pairwise non-isomorphic quasi-ordered difference fields of cardinality $κ$, but all isomorphic as quasi-ordered fields.

math.LO↗