Model Theory of Local Real Closed SV-Rings of Finite Rank

This note begins the model-theoretic study of local real closed SV-rings of finite rank; to this end, a structure theorem for reduced local SV-rings of finite rank is given and branching ideals in local real closed rings of finite rank are analysed. The class of local real closed SV-rings of rank $n\in {\mathbb{N}}^{\ge 2}$ is elementary in the language of rings $\mathcal{L}:=\{+,-,\cdot ,0,1\}$ and its $\mathcal{L}$-theory $T_n$ has a model companion $T_{n,1}$; models of $T_{n,1}$ are $n$-fold fibre products $(((V_1{\times }_{\mathbf{\text{k}}}{V}_2){\times }_{\mathbf{\text{k}}}{V}_3)\cdots {\times }_{\mathbf{\text{k}}}{V}_{n-1}){\times }_{\mathbf{\text{k}}}{V}_n$ of non-trivial real closed valuation rings $V_i$ with isomorphic residue field $\mathbf{k}$. The $\mathcal{L}$-theory $T_{n,1}$ is complete, decidable, and NIP. After enriching $\mathcal{L}$ with a predicate for the maximal ideal, models of $T_n$ have prime extensions in models of $T_{n,1}$, and $T_{n,1}$ is the model completion of $T_n$ in this enriched language. A quantifier elimination result for $T_{n,1}$ is also given. The class of those local real closed SV-rings of rank $n\in {\mathbb{N}}^{\ge 2}$ which are $n$-fold fibre products $(((V_1{\times }_W{V}_2){\times }_W{V}_3)\cdots {\times }_W{V}_{n-1}){\times }_W{V}_n$ of non-trivial real closed valuation rings $V_i$ along surjective morphisms $V_i\twoheadrightarrow W$ onto a non-trivial domain $W$ is elementary in the language of rings, and its $\mathcal{L}$-theory $T_{n,2}$ is also complete, decidable, and NIP; after enriching $\mathcal{L}$ with predicates for the maximal ideal and the unique branching ideal, $T_{n,2}$ is model complete.