We study the cone 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 𝒞_n,2d. We prove that every square of an arbitrary polynomial belongs to the mean polynomial preprime T_mean, that T_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 R̂ as a separating example that lies in ℳ_4,4 but outside 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.

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 multiplication 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 the union of the SOS and SONC cones. This follows by exploiting a new necessary condition for membership in the SONC cone.

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.

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.

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 and the value set of its natural valuation. We consider in particular the case of an almost real closed field and give conditions under which the definable ranks on the level of field, of value group and of value set coincide.

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.

For n, d is an element of N, the cone P-n+1,P-2d of positive semidefinite (PSD) (n+1)-ary 2d-ic forms (i.e., homogeneous polynomials with real coefficients in n + 1 variables of degree 2d) contain the cone Sigma(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 Sigma(n+1,2d) = P-n+1,P-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 examined in a preceeding paper a specific cone filtration Sigma(n+1,2d) = C-0 subset of . . . subset of C-n subset of Cn+1 subset of . . . subset of C-k(n,C-d)-n = P-n+1,P-2d along k(n, d) + 1-n projective varieties containing the Veronese variety via the Gram matrix method. Here, k(n, d) + 1 was the dimension of the real vector space of (n + 1)-ary d-ic forms. In particular, we computed the number mu(n, d) of strictly separating intermediate cones (i.e., C-i such that Sigma(n+1,2d) subset of C-i subset of P-n+1,P-2d) for the cases (3,6) and (n + 1, 2d)(n>3,d=2,3). In this paper, first, we generalize these findings to any non-Hilbert case by identifying each strict inclusion in the above specific cone filtration. This allows us to give a refinement of Hilbert's 1888 theorem by computing mu(n, d) in any non-Hilbert case. The cone filtration thus reduces to a specific cone subfiltration Sigma(n+1,2d) = C'(0) subset of C'(1) subset of . . . subset of C'(mu(n,d))subset of C'(mu(n,d)+1)= P-n+1,P-2d in which each inclusion is strict. Second, we show that each C'(i) , and hence each strictly separating C-i, fails to be a spectrahedral shadow.

In 1888, Hilbert proved that the cone 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≥1 or (n+1,2)_n≥1 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 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 ⊆…⊆ C_n ⊆ C_n+1⊆…⊆ C_k(n,d)-n = 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.

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.

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.

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.

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 $\Sigma_{n+1,2d}$ 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 $\Sigma_{n+1,2d}\subsetneq\mathcal{P}_{n+1,2d}$ 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 $\Sigma_{n+1,2d}$ 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 $\Sigma_{n+1,2d}$. 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 $\Sigma_{n+1,2d}$. 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.

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.

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.

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.

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.

We investigate when a linear functional L defined on a linear sub-space B of a unital commutative real algebra A admits an integral representa-tion with respect to 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 results on truncated moment problems.

This workshop was dedicated to the newest developments in real algebraic geometry and their interaction with convex optimization and operator theory. A particular effort was invested in exploring the interrelations with the Koopman operator methods in dynamical systems and their applications. The presence of researchers from different scientific communities enabled an interesting dialogue leading to new exciting and promising synergies.

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.

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.