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Multiplicative Valued Difference Fields

The theory of valued difference fields $(K, σ, v)$ depends on how the valuation $v$ interacts with the automorphism $σ$. Two special cases have already been worked out - the isometric case, where $v(σ(x)) = v(x)$ for all $x\in K$, has been worked out by Luc Belair, Angus Macintyre and Thomas Scanlon; and the contractive case, where $v(σ(x)) > n\cdot v(x)$ 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 $v(σ(x)) = ρ\cdot v(x)$, where $ρ(> 0)$ 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.