Multiplicative Valued Difference Fields

Details Multiplicative Valued Difference Fields Multiplicative Valued Difference Fields

2010-11-07

arxiv.org/abs/1011.1655v2

Mathematics

Logic

37 pages

Scientific paper

The theory of valued difference fields $(K, \sigma, v)$ depends on how the valuation $v$ interacts with the automorphism $\sigma$. Two special cases have already been worked out - the isometric case, where $v(\sigma(x)) = v(x)$ for all $x\in K$, has been worked out by Luc Belair, Angus Macintyre and Thomas Scanlon; and the contractive case, where $v(\sigma(x)) > n\cdot v(x)$ for all $n\in\mathbb{N}$ and $x\in K^\times$ with $v(x) > 0$, has been worked out by Salih Azgin. In this paper we deal with a more general version, called the multiplicative case, where $v(\sigma(x)) = \rho\cdot v(x)$, where $\rho (> 0)$ is interpreted as an element of a real-closed field. We give an axiomatization and prove a relative quantifier elimination theorem for such a theory.

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