Henselian residually pp-adically closed fields

In [R. Farré, A positivstellensatz for chain-closed fields R((t))R((t)) and some related fields, Archiv der Mathematik 57 (1991), 446–455], R. Farré proved a positivstellensatz for real-series closed fields. Here we consider pp-valued fields 〈K,vp〉〈K,vp〉 with a non-trivial valuation vv which satisfies a compatibility condition between vpvp and vv. We use this notion to establish the pp-adic analogue of real-series closed fields; these fields are called henselian residually  pp-adically closed fields  . First we solve a Hilbert’s Seventeenth problem for these fields and then we introduce the notions of residually pp-adic ideal and residually pp-adic radical of an ideal in the ring of polynomials in nn indeterminates over a henselian residually pp-adically closed field. Thanks to these two notions, we prove a Nullstellensatz theorem for this class of valued fields. We finish the paper with the study of the differential analogue of henselian residually pp-adically closed fields. In particular, we give a solution to a Hilbert’s Seventeenth problem in this setting.