The valuation difference rank of a quasi-ordered difference field

There are several equivalent characterizations of the valuation rank of an ordered or valued field. In this paper, we extend the theory to the case of an ordered or valued {\it difference} field (that is, ordered or valued field endowed with a compatible field automorphism). We introduce the notion of {\it difference rank}. To treat simultaneously the cases of ordered and valued fields, we consider quasi-ordered fields. We characterize the difference rank as the quotient modulo the equivalence relation naturally induced by the automorphism (which encodes its growth rate). In analogy to the theory of convex valuations, we prove that any linearly ordered set can be realized as the difference rank of a maximally valued quasi-ordered difference field. As an application, we show that for every regular uncountable cardinal $\kappa$ such that $\kappa= \kappa^{< \kappa}$, there are $2^{\kappa}$ pairwise non-isomorphic quasi-ordered difference fields of cardinality $\kappa$, but all isomorphic as quasi-ordered fields.