A Nullstellensatz for the field of complex Laurent series

Abstract

Let $T$ be the theory of the valued field of complex Laurent series $\mathbb{C}((t))$ in the language of rings expanded by a constant for $t$ and by predicates for $n$-th powers. We introduce a $t$-adic analogue of Kochen's operator, namely $\gamma (x)=x/(1+t{x}^2)$, and the associated Kochen ring ${\Lambda }_L$ of a field extension $L$ of a model $K$ of $T$. Our main result characterizes the field extensions of $K$ which embed into a model of $T$ (the formally $T$-adic} fields): $L$ is formally $T$-adic if and only if $1/t\notin {\Lambda }_L$. As an application we prove a Nullstellensatz for $\mathbb{C}((t))$: a polynomial $f\in K[X_1,\dots ,X_n]$ vanishes on the zero set of an ideal $I$ if and only if $f^m={\sum }_i{\omega }_i{f}_i$ for some $m⩾1$, generators $f_i$ of $I$ and coefficients ${\omega }_i$ in the ring ${\Lambda }_{K(X)}\cdot K[X]$. We derive the resulting correspondence between $R$-ideals and algebraic sets, and we compare natural incarnations of the associated $T$-spectrum: prime ideals, morphisms into models of $T$, and the space of types of $T$.

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