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 ∈ K, has been worked out by Luc Belair, Angus Macintyre and Thomas Scanlon; and the contractive case, where v(σ(x)) > nv(x) for all x ∈ K× with v(x) > 0 and n ∈ N, has been worked out by Salih Azgin. In this paper we deal with a more general version, the multiplicative case, where v(σ(x)) = ρ·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 this theory.