\(p\)-adic ideals of \(p\)-rank \(d\) and the \(p\)-adic Nullstellensatz. (Q1873798)

The author proposes \(p\)-adic analogues of the notions of real ideal and real radical of an ideal in the real case. So \(p\)-adic ideals and \(p\)-adic radicals of ideals are introduced in rings of polynomials over \(p\)-adically closed fields of a given \(p\)-rank for every prime \(p\). This leads to a new proof of the \(p\)-adic Nullstellensatz, based on the approach of [\textit{J. Bochnak, M. Coste} and \textit{M.-F. Roy}, Géometrie Algebrique Réellel. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Bd. 12, (Berlin etc.: Springer-Verlag). (1987; Zbl 0633.14016)] to the real Nullstellensatz.