Salma Kuhlmann

Positivity, sums of squares and the multi-dimensional moment problem

Let K be the basic closed semi-algebraic set in R n defined by some finite set of polynomials S and T, the preordering generated by S. For K compact, f a polynomial in n variables nonnegative on K and real e > 0, we have that f + ∈ E T. In particular, the K-Moment Problem has a positive solution. In the present paper, we study the problem when K is not compact. For n = 1, we show that the K-Moment Problem has a positive solution if and only if S is the natural description of K (see Section 1). For n > 2, we show that the K-Moment Problem fails if K contains a cone of dimension 2. On the other hand, we show that if K is a cylinder with compact base, then the following property holds: (+) ∀ f ∈ R[X], f ≥ 0 on K ⇒ ∃q ∈ T such that ∀ real ∈ > 0, f + eq ∈ T. This property is strictly weaker than the one given in Schmudgen (1991), but in turn it implies a positive solution to the K-Moment Problem. Using results of Marshall (2001), we provide many (noncompact) examples in hypersurfaces for which () holds. Finally, we provide a list of 8 open problems.

Exponentiation in power series fields

We prove that for no nontrivial ordered abelian group G, the ordered power series fleld R((G)) admits an exponential, i.e. an isomorphism between its ordered additive group and its ordered multiplicative group of positive elements, but that there is a non- surjective logarithm. For an arbitrary ordered fleld k, no exponen- tial on k((G)) is compatible, that is, induces an exponential on k through the residue map. This is proved by showing that certain functional equations for lexicographic powers of ordered sets are not solvable.

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.

Embedding ordered fields in formal power series fields

In the paper it is shown how an embedding of an ordered field F into a formal power series field can be extended canonically to an embedding of any simple extension F(y) of F. Properties of the extended embedding are studied in detail. Several applications are given.

Isomorphisms of lexicographic powers of the reals

Given a chain F, we consider the lexicographic order llr. If A is a chain such that Rr IRA, we examine the question of whether necessarily F A. Under an additional hypothesis, we show that F and A will have the same order types of well ordered subsets. Among other things, this yields an affirmative answer to the above question in the case where F and A are ordinals.

The moment problem for continuous positive semidefinite linear functionals

Let τ be a locally convex topology on the countable dimensional polynomial

Valuation Bases for Extensions of Valued Vector Spaces

{\mathbb{R}}

Hardy type derivations on fields of exponential logarithmic series

-algebra