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

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.