0-D-valued fields

Abstract

In [12]. 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 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 byproduct, we state and prove a “positivstellensatz” theorem for the differential analogue of the theory of real-series closed fields in the valued D-field setting.

Model Theory of Fields With Operators – a Survey

“…Valued differential fields have been studied extensively by Guzy, Michaux, Point and Rivière in [13], [30], [31], [32], [33], [35], [36]. ∀x Dx = 0 → ∃y y p = x.…”

Henselian residually p-adically closed fields

“…Now we apply Corollary 3.14 of [8] in order to prove a model completeness result for the theory of henselian residually p-adically closed D-fields which can be expressed in the first-order language L D, p,a := L p,a ∪ {D}. We denote this L D, p,a -theory by H RpC D F. This model-theoretic result will be needed in the proof of Theorem 4.4 which is a differential Hilbert's Seventeenth problem for henselian residually p-adically closed D-fields.…”

“…Finally, in the last section, we study a special class of D-henselian valued fields (first considered in [15]) which uses the results of [6,8]. In [6], we established and axiomatized the model companion of the theory of differential p-valued fields which is denoted by pC D F and whose models are called p-adically closed differential fields.…”

“…In [6], we established and axiomatized the model companion of the theory of differential p-valued fields which is denoted by pC D F and whose models are called p-adically closed differential fields. It is a p-adic adaptation of the theory of closed ordered differential field (see [17]) which is denoted by C O D F. In [8], we study D-henselian valued fields with residue differential field which is a model of C O D F and with a Z-group as value group, i.e. a differential analogue of the theory of real-series closed fields.…”

“…we are interested in the valued D-field analogue of H RpC F. So we prove a Hilbert's Seventeenth problem for D-henselian valued fields whose residue field is a model of pC D F and whose value group is a Z-group. The model-theoretic tool that we need is a theorem of quantifier elimination in [8]; it enables us to prove the model completeness of the theory of these D-henselian valued fields in a suitable language by using linear divisibility predicates.…”