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Salma Kuhlmann

Publications and source records attributed to Salma Kuhlmann.

At least 55 records · Page 3Linked to original sources
arXiv2016

On the Value Group of a Model of Peano Arithmetic

We investigate $IPA$ - real closed fields, that is, real closed fields which admit an integer part whose non-negative cone is a model of Peano Arithmetic. We show that the value group of an $IPA$ - real closed field is an exponential group in the residue field, and that the converse fails in general. As an application, we classify (up to isomorphism) value groups of countable recursively saturated exponential real closed fields. We exploit this characterization to construct countable exponential real closed fields which are not $IPA$ - real closed fields.

math.LO↗
arXiv2015

A note on $\aleph_α$-saturated o-minimal expansions of real closed fields

We give necessary and sufficient conditions for a polynomially bounded o-minimal expansion of a real closed field (in a language of arbitrary cardinality) to be $\aleph_α$-saturated. The conditions are in terms of the value group, residue field, and pseudo- Cauchy sequences of the natural valuation on the real closed field. This is achieved by an analysis of types, leading to the trichotomy. Our characterization provides a construction method for saturated models, using fields of generalized power series.

math.LO↗
arXiv2015

On the Choi-Lam analogue of Hilbert's 1888 theorem for Symmetric Forms

A famous theorem of Hilbert from 1888 states 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. In 1976, Choi and Lam proved the analogue of Hilbert's Theorem for symmetric forms by assuming the existence of psd not sos symmetric $n$-ary quartics for $n \geq 5$. In this paper we complete their proof by constructing explicit psd not sos symmetric $n$-ary quartics for $n \geq 5$.

math.AG↗
arXiv2014

Moment problem in infinitely many variables

The multivariate moment problem is investigated in the general context of the polynomial algebra $\mathbb{R}[x_i \mid i \in Ω]$ in an arbitrary number of variables $x_i$, $i\in Ω$. The results obtained are sharpest when the index set $Ω$ is countable. Extensions of Haviland's theorem [Amer. J. Math., 58 (1936) 164-168] and Nussbaum's theorem [Ark. Math., 6 (1965) 179-191] are proved. Lasserre's description of the support of the measure in terms of the non-negativity of the linear functional on a quadratic module of $\mathbb{R}[x_i \mid i \in Ω]$ in [Trans. Amer. Math. Soc., 365 (2013) 2489-2504] is shown to remain valid in this more general situation. The main tool used in the paper is an extension of the localization method developed by the third author.

math.FA↗
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.

math.AC↗
arXiv2013

A Note on Schanuel's Conjectures for Exponential Logarithmic Power Series Fields

In [1], J. Ax proved a transcendency theorem for certain differential fields of characteristic zero: the differential counterpart of the still open Schanuel's conjecture about the exponential function over the field of complex numbers [11, page 30]. In this article, we derive from Ax's theorem transcendency results in the context of differential valued exponential fields. In particular, we obtain results for exponential Hardy fields, Logarithmic-Exponential power series fields and Exponential-Logarithmic power series fields.

math.LO↗
arXiv2012

Application of Jacobi's Representation Theorem to locally multiplicatively convex topological real Algebras

Let $A$ be a commutative unital $\mathbb{R}$-algebra and let $ρ$ be a seminorm on $A$ which satisfies $ρ(ab)\leqρ(a)ρ(b)$. We apply T. Jacobi's representation theorem to determine the closure of a $\sum A^{2d}$-module $S$ of $A$ in the topology induced by $ρ$, for any integer $d\ge1$. We show that this closure is exactly the set of all elements $a\in A$ such that $α(a)\ge0$ for every $ρ$-continuous $\mathbb{R}$-algebra homomorphism $α: A \rightarrow \mathbb{R}$ with $α(S)\subseteq[0,\infty)$, and that this result continues to hold when $ρ$ is replaced by any locally multiplicatively convex topology $τ$ on $A$. We obtain a representation of any linear functional $L : A \rightarrow \reals$ which is continuous with respect to any such $ρ$ or $τ$ and non-negative on $S$ as integration with respect to a unique Radon measure on the space of all real valued $\reals$-algebra homomorphisms on $A$, and we characterize the support of the measure obtained in this way.

math.FA↗
arXiv2012

Closure of the cone of sums of 2d-powers in real topological algebras

Let $R$ be a unitary commutative real algebra and $K\subseteq Hom(R,\mathbb{R})$, closed with respect to the product topology. We consider $R$ endowed with the topology $\mathcal{T}_K$, induced by the family of seminorms $ρ_α(a):=|α(a)|$, for $α\in K$ and $a\in R$. In case $K$ is compact, we also consider the topology induced by $\|a\|_K:=\sup_{α\in K}|α(a)|$ for $a\in R$. If $K$ is Zariski dense, then those topologies are Hausdorff. In this paper we prove that the closure of the cone of sums of 2d-powers, $\sum R^{2d}$, with respect to those two topologies is equal to $Psd(K):=\{a\in R:α(a)\geq 0,\textrm{for all}α\in K\}$. In particular, any continuous linear functional $L$ on the polynomial ring $R=\mathbb{R}[X_1,...,X_n]$ with $L(h^{2d})\ge0$ for each $h\in R$ is integration with respect to a positive Borel measure supported on $K$. Finally we give necessary and sufficient conditions to ensure the continuity of a linear functional with respect to those two topologies.

math.RA↗
arXiv2012

The Moment Problem for Continuous Positive Semidefinite Linear functionals

Let $τ$ be a locally convex topology on the countable dimensional polynomial $\reals$-algebra $\rx:=\reals[X_1,...,X_n]$. Let $K$ be a closed subset of $\reals^n$, and let $M:=M_{\{g_1, ... g_s\}}$ be a finitely generated quadratic module in $\rx$. We investigate the following question: When is the cone $\Pos(K)$ (of polynomials nonnegative on $K$) included in the closure of $M$? We give an interpretation of this inclusion with respect to representing continuous linear functionals by measures. We discuss several examples; we compute the closure of $M=\sos$ with respect to weighted norm-$p$ topologies. We show that this closure coincides with the cone $\Pos(K)$ where $K$ is a certain convex compact polyhedron.

math.AG↗
arXiv2012

The exponential-logarithmic equivalence classes of surreal numbers

In his monograph, H. Gonshor showed that Conway's real closed field of surreal numbers carries an exponential and logarithmic map. Subsequently, L. van den Dries and P. Ehrlich showed that it is a model of the elementary theory of the field of real numbers with the exponential function. In this paper, we give a complete description of the exponential equivalence classes in the spirit of the classical Archimedean and multiplicative equivalence classes. This description is made in terms of a recursive formula as well as a sign sequence formula for the family of representatives of minimal length of these exponential classes.

math.AC↗
arXiv2012

Hardy type derivations on generalized series fields

We consider the valued field $\mathds{K}:=\mathbb{R}((Γ))$ of generalized series (with real coefficients and monomials in a totally ordered multiplicative group $Γ$). We investigate how to endow $\mathds{K}$ with a series derivation, that is a derivation that satisfies some natural properties such as commuting with infinite sums (strong linearity) and (an infinite version of) Leibniz rule. We characterize when such a derivation is of Hardy type, that is, when it behaves like differentiation of germs of real valued functions in a Hardy field. We provide a necessary and sufficent condition for a series derivation of Hardy type to be surjective.

math.AC↗
arXiv2011

Comparison of Exponential-Logarithmic and Logarithmic-Exponential series

We explain how the field of logarithmic-exponential series constructed in \cite{DMM1} and \cite {DMM2} embeds as an exponential field in any field of exponential-logarithmic series constructed in \cite{KK1}, \cite {K} and \cite {KS}. On the other hand, we explain why no field of exponential-logarithmic series embeds in the field of logarithmic-exponential series. This clarifies why the two constructions are intrinsically different, in the sense that they produce non-isomorphic models of Th$(\R_{an, exp})$; the elementary theory of the ordered field of real numbers, with the exponential function and restricted analytic functions.

math.LO↗
arXiv2011

Real closed exponential fields

In an extended abstract Ressayre considered real closed exponential fields and integer parts that respect the exponential function. He outlined a proof that every real closed exponential field has an exponential integer part. In the present paper, we give a detailed account of Ressayre's construction, which becomes canonical once we fix the real closed exponential field, a residue field section, and a well ordering of the field. The procedure is constructible over these objects; each step looks effective, but may require many steps. We produce an example of an exponential field $R$ with a residue field $k$ and a well ordering $<$ such that $D^c(R)$ is low and $k$ and $<$ are $Δ^0_3$, and Ressayre's construction cannot be completed in $L_{ω_1^{CK}}$.

math.LO↗
arXiv2011

Hardy type derivations on fields of exponential logarithmic series

We consider the valued field $\mathds{K}:=\mathbb{R}((Γ))$ of formal series (with real coefficients and monomials in a totally ordered multiplicative group $Γ>$). We investigate how to endow $\mathds{K}$ with a logarithm $l$, which satisfies some natural properties such as commuting with infinite products of monomials. In the article "Hardy type derivations on generalized series fields", we study derivations on $\mathds{K}$. Here, we investigate compatibility conditions between the logarithm and the derivation, i.e. when the logarithmic derivative is the derivative of the logarithm. We analyse sufficient conditions on a given derivation to construct a compatible logarithm via integration of logarithmic derivatives. In her monograph "Ordered exponential fields", the first author described the exponential closure $\mathds{K}^{\rm{EL}}$ of $(\mathds{K},l)$. Here we show how to extend such a log-compatible derivation on $\mathds{K}$ to $\mathds{K}^{\rm{EL}}$.

math.AC↗
arXiv2008

Sums of squares and moment problems in equivariant situations

We begin a systematic study of positivity and moment problems in an equivariant setting. Given a reductive group $G$ over $\R$ acting on an affine $\R$-variety $V$, we consider the induced dual action on the coordinate ring $\R[V]$ and on the linear dual space of $\R[V]$. In this setting, given an invariant closed semialgebraic subset $K$ of $V(\R)$, we study the problem of representation of invariant nonnegative polynomials on $K$ by invariant sums of squares, and the closely related problem of representation of invariant linear functionals on $\R[V]$ by invariant measures supported on $K$. To this end, we analyse the relation between quadratic modules of $\R[V]$ and associated quadratic modules of the (finitely generated) subring $\R[V]^G$ of invariant polynomials. We apply our results to investigate the finite solvability of an equivariant version of the multidimensional $K$-moment problem. Most of our results are specific to the case where the group $G(\R)$ is compact.

math.AG↗
arXiv2008

Positivity in power series rings

We extend and generalize the results of Scheiderer (2006) on the representation of polynomials nonnegative on two-dimensional basic closed semialgebraic sets. Our extension covers some situations where the defining polynomials do not satisfy the transversality condition. Such situations arise naturally when one considers semialgebraic sets invariant under finite group actions.

math.AG↗