113 research outputs found
Exponentiation in power series fields
We prove that for no nontrivial ordered abelian group G, the ordered power
series field 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 field k, no exponential 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
A note on -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 -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.Comment: Key words and phrases. natural valuation, value group, residue field,
pseudo- Cauchy sequences, polynomially bounded o-minimal expansion of a real
closed field, definable closure, dimension, saturation. To appear in Algebra
and Logic volume 54 Issue 5 November 201
kappa-bounded Exponential-Logarithmic Power Series Fields
In math.AC/9608214 it was shown that fields of generalized power series
cannot admit an exponential function. In this paper, we construct fields of
generalized power series with bounded support which admit an exponential. We
give a natural definition of an exponential, which makes these fields into
models of real exponentiation. The method allows to construct for every kappa
regular uncountable cardinal, 2^{kappa} pairwise non-isomorphic models of real
exponentiation (of cardinality kappa), but all isomorphic as ordered fields.
Indeed, the 2^{kappa} exponentials constructed have pairwise distinct growth
rates. This method relies on constructing lexicographic chains with many
automorphisms
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
Infinite dimensional moment problem: open questions and applications
Infinite dimensional moment problems have a long history in diverse applied
areas dealing with the analysis of complex systems but progress is hindered by
the lack of a general understanding of the mathematical structure behind them.
Therefore, such problems have recently got great attention in real algebraic
geometry also because of their deep connection to the finite dimensional case.
In particular, our most recent collaboration with Murray Marshall and Mehdi
Ghasemi about the infinite dimensional moment problem on symmetric algebras of
locally convex spaces revealed intriguing questions and relations between real
algebraic geometry, functional and harmonic analysis. Motivated by this
promising interaction, the principal goal of this paper is to identify the main
current challenges in the theory of the infinite dimensional moment problem and
to highlight their impact in applied areas. The last advances achieved in this
emerging field and briefly reviewed throughout this paper led us to several
open questions which we outline here.Comment: 14 pages, minor revisions according to referee's comments, updated
reference
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