Henselian residually p-adically closed fields

In [R. Farré, A positivstellensatz for chain-closed fields image and some related fields, Archiv der Mathematik 57 (1991), 446–455], R. Farré proved a positivstellensatz for real-series closed fields. Here we consider p-valued fields left angle bracketK,vpright-pointing angle bracket with a non-trivial valuation v which satisfies a compatibility condition between vp and v. We use this notion to establish the p-adic analogue of real-series closed fields; these fields are called henselian residually p-adically closed fields. First we solve a Hilbert’s Seventeenth problem for these fields and then we introduce the notions of residually p-adic ideal and residually p-adic radical of an ideal in the ring of polynomials in n indeterminates over a henselian residually p-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 p-adically closed fields. In particular, we give a solution to a Hilbert’s Seventeenth problem in this setting.