In his 1997 thesis, Thomas Scanlon developed the model theory of a class of valued fields, which allow for the consideration of a difference field and a related differential field in the same structure. In this theory, fields are endowed with a derivative like operator D, interacting strongly with a valuation. The operator specializes to a derivative in the residue field, but in the valued field is interdefinable with a nontrivial automorphism. The theory was shown to have good model theoretic properties, most notably quantifier elimination.

We look at solutions to linear D-equations in these fields, with the goal of using the residue differential field to better understand the behavior of the difference field solutions. First, we show that the dimension of a maximal solution space to such an equation as a vector space over the constants is completely determined by the structure induced on the residue field. We then find reasonable conditions on the base field sufficient to assure uniqueness for the field extension generated by these solutions. Finally, we provide examples of automorphism groups in the theory; in particular, we show that nonlinear relations in the residue field may not lift to the valued field.