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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 ‐adic power series (both formal and convergent) analogous to Rückert's complex and Risler's real Nullstellensatz, as well as a ‐adic analytic version of Hilbert's 17th Problem. Analogous statements for restricted power series, both real and ‐adic, are also considered.
Aschenbrenner et al. (Sun,) studied this question.