HENSELIAN RESIDUALLY p-ADICALLY CLOSED FIELDS

Abstract. In (Arch. Math. 57 (1991), pp. 446–455), R. Farre ́ proved a posi-tivstellensatz for real-series closed fields. Here we consider p-valued fields 〈K, vp〉 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