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Abstract: The theory of valued difference fields $K, \sigma, v$ depends on how thevaluation $v$ interacts with the automorphism $\sigma$. Two special cases havealready been worked out - the isometric case, where $v\sigmax = vx$ forall $x\in K$, has been worked out by Luc Belair, Angus Macintyre and ThomasScanlon; and the contractive case, where $v\sigmax > n\cdot vx$ for all$n\in\mathbb{N}$ and $x\in K^\times$ with $vx > 0$, has been worked out bySalih Azgin. In this paper we deal with a more general version, called themultiplicative case, where $v\sigmax = ho\cdot vx$, where $ ho > 0$is interpreted as an element of a real-closed field. We give an axiomatizationand prove a relative quantifier elimination theorem for such a theory.

Author: Koushik Pal



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