Analytic Nullstellensätze and the model theory of valued fields

Matthias Aschenbrenner and Ahmed Srhir

Mathematische Nachrichten, 2024, vol. 297, issue 8, 2873-2917

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

Date: 2024
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https://doi.org/10.1002/mana.202200280

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