Ralph Berr

Positive polynomials on compact sets

Abstract: The aim of this note is to give short algebraic proofs of theorems of Handelman, Pólya and Schmüdgen concerning the algebraic structure of polynomials being positive on certain subsets of ℝn. The main ingredient of the proofs is the representation theorem of Kadison–Dubois. The proof of the latter is elementary and algebraic but tricky.

The intersection theorem for orderings of higher level in rings

This short note is ment as a supplement to the paper “On Rings admitting Orderings and 2-primary Orderings of Higher Level” by E. Becker and D. Gondard ([4]), where an intersection theorem for 2-primary orderings of higher level has been proved ([4]), Proposition 2.6). We will show that the same characterization holds for orderings of arbitrary level. This result finds several applications. For example, it is useful for the continuous representation of sums of 2n-th powers in function fields (see [8]) and it can be applied to derive several Null- and Positivstellensätze for generalized real closed fields (see [5]). As a further example we will prove a strict “Positivstellensatz of higher level” for a certain class of formally real fields. For unexplained notions we refer the reader to [4].

Positive polynomials and tame preorderings

From the geometric point of view it is one of the essential merits of theArtinSchreier theory of ordered fields to have shed new light on certain problems concerning the behaviour of regular functions on an irreducible variety V over R. This eventually led to a purely algebraic characterization of those f ∈ R[V ] which are positive or nonnegative on V (R)reg. A first example is provided by E. Artin’s positive solution of Hilbert’s 17-th problem. In the seventies classical Artin-Schreier theory was extended to commutative rings and generalizations of Artin’s result were proven. For the purpose of this introduction we consider only a special situation. Let us fix polynomials f1, . . . , fk ∈ R[X] = R[X1, . . . , Xn],

Null- and Positivstellensa¨tze for generalized real closed fields

Abstract Null- and Positivstellensa¨tze for all generalized real closed fields are proved. These fields play an important role for the study of sums of powers in fields. The results of this paper are especially related to the study of sums of powers in real function fields.

COMPLETELY REAL HULLS OF PARTIALLY ORDERED RINGS

In this paper we investigate certain extensions of partially ordered rings. This purely algebraic approach is inspired by the search for rings of ‘‘rational functions’’ on semialgebraic sets which should play the same role as the ring of regular functions in the framework of algebraic geometry. Therefore, in order to motivate the consideration of the algebraic constructions below we first briefly describe the geometric background. Let R be a real closed field and let S AðRÞ be a closed semialgebraic set. We let R1⁄2S denote the ring of R-valued polynomial functions on S and P1⁄2S R1⁄2S the partial order of nonnegative functions on S. Then it is known that the irreducible algebraic subsets of S correspond bijectively to the P1⁄2S -convex prime ideals of R1⁄2S . Consequently, in order to study the geometry of algebraic subsets of S it might seem natural to investigate the locally ringed space ðXS;OSÞ, where XS SpecR1⁄2S denotes the proconstructible subspace of the P1⁄2S -convex prime ideals and OS the restriction of the structure sheaf of the affine scheme ðSpecR1⁄2S ;OÞ to XS. From the geometric point of view the space ðXS;OSÞ has some convenient properties. For example, the points of S are in bijective correspondence with the maximal ideals of the ring OSðXSÞ ffi R1⁄2S 1þP1⁄2S of global sections on XS. But there is also the serious problem that the ideal-theoretic structure of the ring OSðXSÞ is only loosely related with the underlying space XS. To be precise,

Sums of powers in function fields

Abstract When is a rational function on a smooth variety V whose values are sums of 2nth powers, itself a sum of 2nth powers in the field of rational functions? We investigate this property and some weaker form of it. The aim is to gain a better understanding of sums of powers in formally real function fields.