Multiplicative Valued Difference Fields

Abstract

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.

37 pages