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A Note on Exponential-Logarithmic Series Fields. ⁄
| Content Provider | Semantic Scholar |
|---|---|
| Author | Kuhlmann, Salma |
| Copyright Year | 2009 |
| Abstract | Let G be an ordered abelian group and R((G)) the field of generalized series. In [KS05] we define exponential functions on the κ-maximal subfield R((G))κ; the subfield of series with support of cardinality bounded by a regular uncountable cardinal κ. Equipped with these exponentials, R((G))κ is a model of Texp; the elementary theory of (R, exp). The purpose of this note is to give a natural functional interpretation of the formal construction of [KS05]. To this end, the group G considered here is the group of transmonomials determined by a totally ordered set Γ of (germs at +∞ of) real valued functions (Section 2.1). More precisely, we construct such a Γ of cardinality א1. We show that Γ is isomorphic to the lexicographic product א1 × Z × Z; which admits 2א1 automorphisms of pairwise distinct orbital growth (Section 3.2). We associate to each such automorphism a well defined logarithmic function on the field R((G))א1 , where R((G))א1 is the field of generalized series with countable support, real coefficients and exponents in the group G defined by Γ (Section 2.1). We show that distinct automorphisms induce logarithmic functions of distinct growth rates (Section 3). In contrast to the notation of [KS05] and [Kuh00], G here is written in multiplicative notation (which is better adapted to the functional setting). For the convenience of the reader, we summarize in multiplicative notation some definitions and results from [KS05] and [Kuh00] (Section 2). |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://math.usask.ca/~skuhlman/KuhlmannRolin(7)2009.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
