For an arbitrary valued field (K,v) and a given extension v(K⁎)↪Λ of ordered groups, we analyze the structure tree formed by all Λ-valued extensions v to polynomial ring K[x]. As application, find model for equivalence classes valuations on K[x] (without fixing their value group), whose restriction K is equivalent v.

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 b...

In [Sca99], T. Scanlon proved a quantifier elimination result for valued D-fields in a three-sorted language by using angular component functions. Here we prove an analogous theorem in a different language L2 which was introduced by F. Delon in her thesis. This language allows us to lift the quantifier elimination result to a one-sorted language by a process described in the Appendix. As a bypr...