Multiplicative valued difference fields

The theory of valued difference fields (K, σ, υ,) depends on how the valuation υ interacts with the automorphism σ. Two special cases have already been worked out - the isometric case, where υ(σ(x)) = υ(x) for all x Є K, has been worked out by Luc Belair, Angus Macintyre and Thomas Scanlon; and the contractive case, where υ(σ(x)) > nυ(x) for all x Є K× with υ(x) > 0 and n Є ℕ, has been worked out by Salih Azgin. In this paper we deal with a more general version, the multiplicative case, where υ(σ(x)) = ρ · υ(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.