arXiv subjects

Salma Kuhlmann

Publications and source records attributed to Salma Kuhlmann.

At least 19 recordsLinked to original sources

arXiv2026

The Cone Generated by Positive Semidefinite Mean Polynomials

We study the cone $\mathcal{M}_{n,2d}$ of nonnegative mean polynomials---real $n$-variate forms of degree $2d$ that can be expressed as weighted power means $M_{q,p}(Y,w)$ with $q>p$. This cone simultaneously generalises the cone of sums of squares $Σ_{n,2d}$ and the cone of sums of nonnegative circuit polynomials $\mathcal{C}_{n,2d}$. We prove that every square of an arbitrary polynomial belongs to the mean polynomial preprime $T_{\mathrm{mean}}$, that $T_{\mathrm{mean}}$ is strongly generating, and consequently that every polynomial strictly positive on a compact semialgebraic set admits a representation with mean polynomial certificates. We exhibit the Robinson form $\hat{R}$ as a separating example that lies in $\mathcal{M}_{4,4}$ but outside $\mathrm{SOSONC}_{4,4}$. Finally, we outline a convergent hierarchy of lower bounds for polynomial optimization based on the mean polynomial cone and discuss tractable depth-truncated approximations via signomial programming.

math.OC↗

arXiv2026

Symmetric tensor decomposition on rational varieties

We study the Waring decomposition of symmetric tensors with nodes on a rational variety. We provide an explicit characterisation of the existence of such a decomposition under some technical assumption, and introduce an efficient algorithm to decompose this novel class of structured symmetric tensors. The framework directly generalizes Hankel tensors (Qi 2015) to the multivariate setting. We analyse in details the case of toric varieties and rational curves. Proving the existence of a quadrature formula of even strength 2N with at most N + 1 nodes, that avoids a prescribed finite set of points, we establish new sharp upper bounds on the minimal number of nodes for quadrature formulae on rational curves. Numerical experimentation demonstrates the gain of this approach, compared to classical direct approaches.

math.AG↗

arXiv2026

Automorphisms of valued Hahn groups

Hahn groups endowed with the canonical valuation play a fundamental role in the classification of valued abelian groups. In this paper we study the group of valuation (respectively order) preserving automorphisms of a Hahn group $G$. Under the assumption that $G$ satisfies some lifting property, we prove a structure theorem decomposing the automorphism group into a semidirect product of two notable subgroups. We characterise a class of Hahn groups satisfying the aforementioned lifting property. For some special cases we provide a matrix description of the automorphism group.

math.GR↗

arXiv2026

Definable ranks

We introduce the notion of the definable rank of an ordered field, ordered abelian group and ordered set, respectively. We study the relation between the definable rank of an ordered field and the definable rank of the value group of its natural valuation. Similarly, we compare the definable rank of an ordered abelian group to that of its value set with respect to the natural valuation. We describe the definable rank on the group-level by characterizing the definable convex subgroups. We also give a detailed comparison of field- and group-level, in particular for ordered fields with henselian natural valuation. We investigate definability of final segments in ordered sets and introduce definable condensation as a tool for further study.

math.LO↗

arXiv2025

Automorphisms and derivations on algebras endowed with formal infinite sums

We establish a correspondence between automorphisms and derivations on certain algebras of generalised power series. In particular, we describe a Lie algebra of derivations on a field $k(\!(G)\!)$ of generalised power series, exploiting our knowledge of its group of valuation preserving automorphisms. The correspondence is given by the formal Taylor expansion of the exponential. In order to define the exponential map, we develop an appropriate notion of summability of infinite families in algebras. We show that there is a large class of algebras in which the exponential induces the above correspondence.

math.RA↗

arXiv2025

Generalised power series determined by linear recurrence relations

In 1882, Kronecker established that a given univariate formal Laurent series over a field can be expressed as a fraction of two univariate polynomials if and only if the coefficients of the series satisfy a linear recurrence relation. We introduce the notion of generalised linear recurrence relations for power series with exponents in an arbitrary ordered abelian group, and generalise Kronecker's original result. In particular, we obtain criteria for determining whether a multivariate formal Laurent series lies in the fraction field of the corresponding polynomial ring. Moreover, we study distinguished algebraic substructures of a power series field, which are determined by generalised linear recurrence relations. In particular, we identify generalised linear recurrence relations that determine power series fields satisfying additional properties which are essential for the study of their automorphism groups.

math.AC↗

arXiv2024

Geometrical Study of the Cone of Sums of Squares plus Sums of Nonnegative Circuits

In this article, we combine sums of squares (SOS) and sums of nonnegative circuit (SONC) forms, two independent nonnegativity certificates for real homogeneous polynomials. We consider the convex cone SOS+SONC of forms that decompose into a sum of an SOS and a SONC form and study it from a geometric point of view. We show that the SOS+SONC cone is proper and neither closed under multiplications nor under linear transformation of variables. Moreover, we present an alternative proof of an analog of Hilbert's 1888 Theorem for the SOS+SONC cone and prove that in the non-Hilbert cases it provides a proper superset of both the SOS and the SONC cone. This follows by exploiting a new necessary condition for membership in the SONC cone.

math.AG↗

arXiv2024

Separating Cones defined by Toric Varieties: Some Properties and Open Problems

In 1888, Hilbert proved that the cone $\mathcal{P}_{n+1,2d}$ of positive semidefinite forms in $n+1$ variables of degree $2d$ coincides with its subcone $Σ_{n+1,2d}$ of those forms that are representable as finite sums of squares if and only if $(n+1,2d) = (2,2d)_{d\geq1}$ or $(n+1,2)_{n\geq1}$ or $(3,4)$. These are the Hilbert cases. In [GHK23, GHK24], we applied the Gram matrix method to construct cones between $Σ_{n+1,2d}$ and $\mathcal{P}_{n+1,2d}$, defined by projective varieties containing the Veronese variety. In particular, we introduced and examined a specific cone filtration $$Σ_{n+1,2d} = C_0 \subseteq \ldots \subseteq C_n \subseteq C_{n+1} \subseteq \ldots \subseteq C_{k(n,d)-n} = \mathcal{P}_{n+1,2d}$$ and determined each strict inclusion in non-Hilbert cases. This gave us a refinement of Hilbert's 1888 theorem. Here, $k(n,d)+1$ is the dimension of the vector space of forms in $n+1$ variables of degree $d$. In this paper, we show that the intermediate cones $C_i$'s are closed and describe their interiors and boundaries. We discuss the membership problem for the $C_i$'s, present open problems concerning their dual cones and generalizations to cones defined by toric varieties.

math.AG↗

arXiv2024

Definable henselian valuations on dp-minimal real fields

We give an explicit algebraic characterisation of all definable henselian valuations on a dp-minimal real field. Additionally we characterise all dp-minimal real fields that admit a definable henselian valuation with real closed residue field. We do so by first proving this for the more general setting of almost real closed fields.

math.LO↗

arXiv2024

On nonnegative invariant quartics in type A

The equivariant nonnegativity versus sums of squares question has been solved for any infinite series of essential reflection groups but type A. As a first step to a classification, we analyse $A_n$-invariant quartics. We prove that the cones of invariant sums of squares and nonnegative forms are equal if and only if the number of variables is at most 3 or odd.

math.AG↗

arXiv2024

The Truncated Moment Problem for Unital Commutative R-Algebras

We investigate when a linear functional $L$ defined on a linear subspace $B$ of a unital commutative real algebra $A$ admits an integral representation w.r.t. a positive Radon measure supported on a closed subset $K$ of the character space of $A$. We provide a criterion for the existence of such a representation for $L$ when $A$ is equipped with a submultiplicative seminorm. We then build on this result to prove our main theorem for $A$ not necessarily equipped with a topology. This allows us to extend well-known classical results on truncated moment problems.

math.FA↗

arXiv2024

A Refinement of Hilbert's 1888 Theorem: Separating Cones along the Veronese Variety

For $n,d\in\mathbb{N}$, the cone $\mathcal{P}_{n+1,2d}$ of positive semi-definite (PSD) $(n+1)$-ary $2d$-ic forms (i.e., homogeneous polynomials with real coefficients in $n+1$ variables of degree $2d$) contains the cone $Σ_{n+1,2d}$ of those that are representable as finite sums of squares (SOS) of $(n+1)$-ary $d$-ic forms. Hilbert's 1888 Theorem states that $Σ_{n+1,2d}=\mathcal{P}_{n+1,2d}$ exactly in the Hilbert cases $(n+1,2d)$ with $n+1=2$ or $2d=2$ or $(3,4)$. For the non-Hilbert cases, we examine in [GHK] a specific cone filtration \begin{equation} Σ_{n+1,2d}=C_0\subseteq \ldots \subseteq C_n \subseteq C_{n+1} \subseteq \ldots \subseteq C_{k(n,d)-n}=\mathcal{P}_{n+1,2d}\end{equation} along $k(n,d)+1-n$ projective varieties containing the Veronese variety via the Gram matrix method. Here, $k(n,d)+1$ is the dimension of the real vector space of $(n+1)$-ary $d$-ic forms. In particular, we compute the number $μ(n,d)$ of strictly separating intermediate cones (i.e., $C_i$ such that $Σ_{n+1,2d}\subsetneq C_i \subsetneq \mathcal{P}_{n+1,2d}$) for the cases $(3,6)$ and $(n+1,2d)_{n\geq 3,d=2,3}$. In this paper, firstly, we generalize our findings from [GHK] to any non-Hilbert case by identifying each strict inclusion in the above cone filtration. This allows us to give a refinement of Hilbert's 1888 Theorem by computing $μ(n,d)$. The above cone filtration thus reduces to a specific cone subfiltration \begin{equation} Σ_{n+1,2d}=C_0^\prime\subsetneq C_1^\prime \subsetneq \ldots \subsetneq C_{μ(n,d)}^\prime \subsetneq C_{μ(n,d)+1}^\prime=\mathcal{P}_{n+1,2d} \end{equation} in which each inclusion is strict. Secondly, we show that each $C_i^\prime$, and hence each strictly separating $C_i$, fails to be a spectrahedral shadow.

math.AG↗

arXiv2023

Ordered transexponential fields

We develop a first-order theory of ordered transexponential fields in the language $\{+,\cdot,0,1,<,e,T\}$, where $e$ and $T$ stand for unary function symbols. While the archimedean models of this theory are readily described, the study of the non-archimedean models leads to a systematic examination of the induced structure on the residue field and the value group under the natural valuation. We establish necessary and sufficient conditions on the value group of an ordered exponential field $(K,e)$ to admit a transexponential function $T$ compatible with $e$. Moreover, we give a full characterisation of all countable ordered transexponential fields in terms of their valuation theoretic invariants.

math.LO↗

arXiv2023

Infinite-dimensional moment-SOS hierarchy for nonlinear partial differential equations

We formulate a class of nonlinear {evolution} partial differential equations (PDEs) as linear optimization problems on moments of positive measures supported on infinite-dimensional vector spaces. Using sums of squares (SOS) representations of polynomials in these spaces, we can prove convergence of a hierarchy of finite-dimensional semidefinite relaxations solving approximately these infinite-dimensional optimization problems. As an illustration, we report on numerical experiments for solving the heat equation subject to a nonlinear perturbation.

math.OC↗

arXiv2023

Intermediate Cones between the Cones of Positive Semidefinite Forms and Sums of Squares

The cone $\mathcal{P}_{n+1,2d}$ ($n,d\in\mathbb{N}$) of all positive semidefinite (PSD) real forms in $n+1$ variables of degree $2d$ contains the subcone $Σ_{n+1,2d}$ of those that are representable as finite sums of squares (SOS) of real forms of half degree $d$. In 1888, Hilbert proved that these cones coincide exactly in the Hilbert cases $(n+1,2d)$ with $n+1=2$ or $2d=2$ or $(n+1,2d)=(3,4)$. To establish the strict inclusion $Σ_{n+1,2d}\subsetneq\mathcal{P}_{n+1,2d}$ in any non-Hilbert case, one can show that verifying the assertion in the basic non-Hilbert cases $(4,4)$ and $(3,6)$ suffices. In this paper, we construct a filtration of intermediate cones between $Σ_{n+1,2d}$ and $\mathcal{P}_{n+1,2d}$. This filtration is induced via the Gram matrix approach (by Choi, Lam and Reznick) on a filtration of irreducible projective varieties $V_{k-n}\subsetneq \ldots \subsetneq V_n \subsetneq \ldots \subsetneq V_0$ containing the Veronese variety. Here, $k$ is the dimension of the vector space of real forms in $n+1$ variables of degree $d$. By showing that $V_0,\ldots,V_n$ are varieties of minimal degree, we demonstrate that the corresponding intermediate cones coincide with $Σ_{n+1,2d}$. Likewise, for the special case when $n=2$, $V_{n+1}$ is also a variety of minimal degree and the corresponding intermediate cone also coincides with $Σ_{n+1,2d}$. We moreover prove that, in the non-Hilbert cases of $(n+1)$-ary quartics for $n\geq 3$ and $(n+1)$-ary sextics for $n\geq 2$, all the remaining cone inclusions are strict.

math.AG↗

arXiv2023

Moment problem for algebras generated by a nuclear space

We establish a criterion for the existence of a representing Radon measure for linear functionals defined on a unital commutative real algebra $A$, which we assume to be generated by a vector space $V$ endowed with a Hilbertian seminorm $q$. Such a general criterion provides representing measures with support contained in the space of characters of $A$ whose restrictions to $V$ are $q-$continuous. This allows us in turn to prove existence results for the case when $V$ is endowed with a nuclear topology. In particular, we apply our findings to the symmetric tensor algebra of a nuclear space.

math.FA↗

arXiv2022

Definable valuations on ordered fields

We study the definability of convex valuations on ordered fields, with a particular focus on the distinguished subclass of henselian valuations. In the setting of ordered fields, one can consider definability both in the language of rings $\mathcal{L}_{\mathrm{r}}$ and in the richer language of ordered rings $\mathcal{L}_{\mathrm{or}}$. We analyse and compare definability in both languages and show the following contrary results: while there are convex valuations that are definable in the language $\mathcal{L}_{\mathrm{or}}$ but not in the language $\mathcal{L}_{\mathrm{r}}$, any $\mathcal{L}_{\mathrm{or}}$-definable henselian valuation is already $\mathcal{L}_{\mathrm{r}}$-definable. To prove the latter, we show that the value group and the ordered residue field of an ordered henselian valued field are stably embedded (as an ordered abelian group, respectively as an ordered field). Moreover, we show that in almost real closed fields any $\mathcal{L}_{\mathrm{or}}$-definable valuation is henselian.

math.LO↗

arXiv2022

Definability of henselian valuations by conditions on the value group

Given a henselian valuation, we study its definability (with and without parameters) by examining conditions on the value group. We show that any henselian valuation whose value group is not closed in its divisible hull is definable in the language of rings, using one parameter. Thereby we strengthen known definability results. Moreover, we show that in this case, one parameter is optimal in the sense that one cannot obtain definability without parameters. To this end, we present a construction method for a $t$-henselian non-henselian ordered field elementarily equivalent to a henselian field with a specified value group.

math.LO↗