Analytic Nullstellens\"atze and the model theory of valued fields

TL;DR

This paper develops a unified approach to Nullstellensatz results in power series rings over valued fields, leveraging model theory and quantifier elimination, with applications to p-adic and real analytic contexts.

Contribution

It introduces a general framework for Nullstellensatz in power series rings using valued field model theory, extending classical results to p-adic and real settings.

Findings

Nullstellensatz for p-adic power series established

Analogues of R"uckert's and Risler's Nullstellensatz obtained

A p-adic version of Hilbert's 17th Problem derived

Abstract

We present a uniform framework for establishing Nullstellens\"atze for power series rings using quantifier elimination results for valued fields. As an application we obtain Nullstellens\"atze for $p$-adic power series (both formal and convergent) analogous to R\"uckert's complex and Risler's real Nullstellensatz, as well as a $p$-adic analytic version of Hilbert's 17th Problem. Analogous statements for restricted power series, both real and $p$-adic, are also considered.