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+\documentclass[amalog]{svjour}
+\usepackage{amsmath}
+\usepackage{amssymb}
+% use Times for text fonts
+%%%%%\usepackage{times}
+% use Times also for mathematics
+%%%%%\usepackage[LY1]{fontenc}
+%%%%%\usepackage[LY1,mtbold]{mathtime}
+%%%%%% coding for first page of a contribution
+%%%%%\input{totalin.aml}
+%%%%%\idline{40: 21--25}{21} % fuer Druckversion
+%\idline{Probab. Theory Relat. Fields 116}{0} % fuer elektronische Fassung (ohne Seitennummern)
+%
+%%%%%\DOIyear{1999} % Jahr der Erstveroeffentlichung ("Online first")
+%%%%%\DOI{123} % Manuskriptnummer
+%
+% Die Jahreszahl fuer die bibliographische Angabe, sowie die im
+% Copyrightvermerk stammen aus dem Maschinendatum. Sie kann mittels
+% \year=2022 % 2022 ist hier nur als Beispiel gewaehlt
+% geaendert werden.
+%%%%%\year=2000
+%
+\def \R { {\bold R} }
+\def \Q { {\bold Q} }
+\def \A {{\bold A}}
+
+\def \F {{\bold F}}
+\def \<{\langle}
+\def \>{\rangle}
+\def \ee{\preceq}
+\def \tilde {\widetilde}
+\def \N {\bold N}
+\def \Z {{\bold Z}}
+\def \E {\text E}
+\def \L {\text L}
+
+\def\(( {(\!(}
+\def\)) {)\!)}
+\def\k{\text{\fontseries{b}\fontshape{it}\selectfont k}}
+
+\def \hat {\widehat}
+\def \Supp {\operatorname{Supp}}
+\def \Suppo {{\Supp}^* }
+\def \iso{\cong}
+
+\def\Cal#1{{\mathcal #1}}
+\def \m{\mathfrak{m}}
+\def \Pos {\operatorname{Pos}}
+\def \supp{\operatorname{supp}}
+\def \bar{\overline}
+
+\spnewtheorem*{thm}{Theorem}{\bf}{\it}
+\spnewtheorem*{prop}{Proposition}{\bf}{\it}
+\spnewtheorem*{rem}{Remark}{\it}{\rm}
+
+\smartqed
+
+\begin{document}
+\title{Riemannian geometry and Hilbert space applied to metamagical
+game theory and the survival problem of Sch\"odinger's
+cat\thanks{K.\,F.\,Gauss was supported by a grant of the Swiss
+National Science Foundation. J.\,H.\,Poincar\'e was partially supported
+by NSF grants DMS-98043192 and NCR-9850637 at Boston University}}
+
+\subtitle{I. Steps towards a theory of almost everything}
+
+\author{K.\,F.\,Gauss \and J.\,H.\,Poincar\'e }
+
+\titlerunning{Elementary theory of rings of Witt vectors}
+
+\institute{K. F. Gauss\at Department of Mathematics, University of
+Illinois at
+Urbana-Champaign,\\ Urbana, IL 61801, USA.
+\email{gauss@math.uiuc.edu}
+\and
+J.\,H.\,Poincar\'e\at Equipe d'Analyse, Tour 46, Universit\'e de Paris
+VI, 4 Place Jussieu,\\ F-75230 Paris Cedex 05, France.
+\email{poincare@math.bu.edu}}
+
+\date{Received: 31 March 1999\,/\,Revised version: 14 October 1999\,/\\
+Published online: 20 December 1999}
+%%%\def\year{2000}
+\subclass{03A60, 12K05, 13L05}
+\keywords{Riemannian geometry -- Hilbert space -- Game theory --
+Survival --\\ Schr\"odinger's cat}
+\maketitle
+\begin{abstract}We indicate a complete set of elementary invariants for
+the ring of Witt vectors over a perfect field of prime characteristic, where this
+ring is equipped with its unique multiplicative set of representatives for
+the residue field.
+\end{abstract}
+
+\noindent
+Theorems of Ax, Kochen and Ersov tell us that the elementary theory of a
+henselian valuation ring of equal
+characteristic $0$ is completely determined
+by the elementary theories of its value group and residue field, see
+\cite{ref1,ref3,ref4}, and the references therein. This elementary
+classification goes through even when
+a predicate is added for a field of representatives of the residue field.
+
+Here we provide a mixed characteristic analogue of the latter when
+the residue field is perfect of characteristic $p$ and the
+maximal ideal is generated by $p$. For a complete discrete valuation ring
+with these properties the analogue of `field of representatives' is
+`multiplicative set of representatives for the residue field' and is due to
+Witt. For proofs of this and related results mentioned below that we shall
+use we refer to Serre \cite[Ch. II]{ref5}.
+We now proceed to precise statements.
+
+Fix a prime number $p$. Let $A$ be a complete discrete valuation ring
+with maximal ideal $\m=pA$ and perfect residue field $\k = A/\m$ (of
+characteristic $p$). Let $\pi : A \to \k$ be the residue class map.
+There is a unique multiplicatively closed set $S \subseteq A$ that is
+mapped bijectively onto $\k$ by $\pi$. (Among the elements of $S$ are
+$0$, $1$ and $-1$.) Each element $a\in A$ can be written uniquely as
+$a=\sum_{i=0}^{\infty}s_ip^i$ with coefficients $s_i\in S$. By \cite[p.
+413]{ref3} one can axiomatize $\operatorname{Th}(A)$ in terms of
+$\operatorname{Th}(\k)$. We extend this to an axiomatization of
+$\operatorname{Th}(A,S)$ in the theorem below. In its proof we shall use
+the functor $W$ that assigns to $\k$ the corresponding ring $W(\k)$ of
+``Witt vectors'' over $\k$. The rings $W(\k)$ and $A$ are isomorphic.
+
+As in \cite[p. 44]{ref5}, let $f : \k \to A$ denote
+the system of multiplicative representatives, that is, $\pi(f(x)) = x$ and $f(xy) = f(x)f(y)$
+for $x,y\in \k$, and thus $f(\k) = S$.
+The following easy result on $\Z$-linear relations among elements of $S$ is decisive.
+
+Let $k = (k_1,\dots,k_n)$ be an $n$-tuple of integers, and let $X=(X_1,\dots,X_n)$ be
+an $n$-tuple of distinct indeterminates. Given an $n$-tuple $b= (b_1,\dots,b_n)$ of elements in an abelian
+(additive) group $B$, put $ k\cdot b := k_1b_1 + \cdots + k_nb_n$.
+
+\begin{lemma}There are polynomials $R_1,\dots,R_N \in \F_p[X]$,
+depending only on
+$p$ and
+ $ k$ and not on $A$, such that for all $ x=(x_1,\dots,x_n) \in \k^n$:
+$$
+k \cdot f(x)= 0 \Longleftrightarrow R_1( x) = \dots = R_N(x) = 0,
+$$
+where $f(x) :=(f(x_1),\dots,f(x_n).$
+\end{lemma}
+\begin{proof}By \cite[Prop. 9, p. 47]{ref5} we have for $x \in \k^n$:
+$$ k \cdot f(x) = \sum_{i=0}^{\infty} f\bigl(P_i( x^{p^{-i}})\bigr)p^i$$
+where $P_i\in \F_p[X]$ depends only on $i$, $p$ and $k$. The ideal of $\F_p[X]$
+generated by the polynomials $P_i$, $i\in \N$, is generated by finitely many among them, say
+$R_1,\dots,R_N$. Then $R_1,\dots,R_N$ have the property described in
+the lemma.\qed\end{proof}
+
+
+The $\Z$-linear relations together with the multiplicative relations $s=s_1s_2$ among the
+elements of $S$ generate all polynomial relations over $\Z$ among elements of $S$:
+
+
+\begin{lemma}Let $U$ and $V$ be multiplicatively closed subsets of
+fields $E$ and $F$
+of characteristic $0$. Let $\lambda : U \to V$ be a bijection such that $\lambda(u_1u_2) =
+\lambda(u_1)\lambda(u_2)$ for all $u_1,u_2\in U$, and such that for all
+$ k \in \Z^n$ and all $u \in U^n$ we have:
+$ k \cdot u = 0 \Longleftrightarrow k \cdot \lambda(u) = 0$.
+Then $\lambda$ extends to an isomorphism from the subfield $\Q(U)$ of $E$ onto the subfield
+$\Q(V)$ of $F$.\end{lemma}
+\begin{proof}Let $P= \sum_i c_iX^i\in \Z[X]$ where the sum is over finitely
+many $i\in \N^n$. Then, given $u\in U^n$, we have $P(u) = \sum_i c_iu^i= 0$ if and only if
+$\sum_i c_i \lambda(u^i) = \sum_i c_i\lambda(u)^i = P(\lambda(u)) = 0$, by the hypothesis of
+the lemma. The conclusion of the lemma follows easily.\qed\end{proof}
+
+ For each $k\in \Z^n$ we fix a tuple $R = (R_1,\dots,R_N)\in \F_p[X]^N$ with the
+property of Lemma 1.
+Let $T$ be the theory in the language
+$\{0,1,+,-,\cdot,{\bold S} \}$ (the language of rings with an extra unary predicate
+${\bold S}$) whose models are the structures $(B,\Sigma)$
+such that \begin{description}[(2)]
+\item[(1)] $B$ is a valuation ring with fraction field $E$ of
+characteristic $0$.
+\item[(2)] $\Sigma$ is a multiplicatively closed subset of $B$ that is
+mapped bijectively
+onto $B/\m(B)$ by the residue class map $b \mapsto \bar b : B \to B/\m(B)$.
+\item[(3)] $\m(B) = pB$ and $B/\m(B)$ is a perfect field.
+\item[(4)] The local ring $B$ is henselian.
+\item[(5)] For each $k\in \Z^n$ we have: $k\cdot \sigma = 0
+\Longleftrightarrow R(\bar \sigma) = 0$,
+for all $\sigma\in \Sigma^n$, where $R\in \F_p[X]^N$ is the tuple associated to $k$, and
+$\bar \sigma := (\bar \sigma_1,\dots,\bar \sigma_n)$.
+\end{description}
+
+\begin{thm} Two models $(B,\Sigma)$ and $(B',\Sigma')$ of $T$ are
+elementarily
+equivalent if and only if their residue fields $B/\m(B)$ and $B'/\m(B')$ are elementarily equivalent,
+and their value groups $\Gamma$ and $\Gamma'$ are elementarily equivalent.
+\end{thm}
+
+Here $\Gamma=v(E^{\times})$ is the value group of the valuation $v$ on the fraction field $E$ of $B$ with valuation ring $B$, and $\Gamma'$, $v'$ and $E'$
+are defined in the same way with $B'$ instead of $B$. These value groups are
+considered as ordered abelian groups.
+
+Following Kochen \cite[pp. 407--408]{ref3}, the idea of the proof is to
+pass to
+sufficiently
+saturated models where the valuation can be decomposed into a valuation of
+equal characteristic $0$ and a complete discrete valuation.
+
+\begin{proof}One direction is obvious. For the other direction we assume that $B/\m(B)
+\equiv B'/\m(B')$ and $\Gamma \equiv \Gamma'$. To show that then $(B,\Sigma) \equiv (B', \Sigma')$, we may assume these two models of $T$ are
+$\aleph _1$-saturated. We focus on $(B,\Sigma)$, but the same analysis will apply
+to $(B',\Sigma')$. We coarsen $v$ to the valuation $\tilde v$
+on $E$ with value group $\tilde \Gamma := \Gamma/\Z\cdot 1$ (where $1:= v(p)$ is the smallest
+positive element of $\Gamma$) by setting $\tilde v(a) = v(a) + \Z\cdot 1$ for $a\in E^{\times}$.
+The valuation ring of $\tilde v$ is
+$$\tilde B := B[1/p] = \{a\in E: v(a) \ge -n\cdot 1 \text{ for some } n\}$$ with
+maximal ideal $\tilde \m := \m(\tilde B) = \{a\in E: v(a) \ge n\cdot 1 \text{ for all } n\}$, and
+residue field $K:= \tilde B/\tilde \m$ of characteristic $0$. Then $\tilde \m$ is also
+a prime ideal of $B$, and $A:= B/\tilde \m$ is a valuation ring of $K$, with maximal ideal
+$pA$. The residue class map $\lambda : \tilde B\to K$ maps $B$ onto $A$, and induces by passing to
+quotients an isomorphism $B/pB \cong A/pA$ of the residue fields of $B$ and $A$. We put
+$\k := B/pB = A/pA$ by identifying these residue fields
+via this isomorphism. Thus $p=\pi \circ (\lambda|B)$ where $p: B \to \k$ and
+$\pi : A \to \k$ are the residue class maps. Hence $S:= \lambda(\Sigma)$ is a multiplicatively
+closed subset of $A$ that is mapped bijectively onto $\k$ by the residue class map $A \to \k$.
+By $\aleph_1$-saturation $A$ is a complete discrete valuation ring, and therefore $(A,S)$ is
+also a model of $T$, by Lemma 1. We now show how to ``lift'' the
+quotient $(K,A,S)$ of
+$(\tilde B,B,\Sigma)$ back to
+$(\tilde B,B,\Sigma)$. The bijection $\sigma \mapsto \lambda(\sigma) : \Sigma \to S$ is
+multiplicative, so by the second lemma $\lambda$ maps the ring $\Z[\Sigma]$ isomorphically
+onto $\Z[S] \subseteq K$. Thus the fraction field $\Q(\Sigma)\subseteq E$ of $\Z[\Sigma]$
+is actually contained in $\tilde B$, and
+$\lambda$ maps $\Q(\Sigma)$ isomorphically onto $\Q(S)$. Since $B$ is henselian, so is its
+localization $\tilde B$. The residue field $K$ of $\tilde B$ being of characteristic $0$,
+it follows that there is a field $L$ with $\Q(\Sigma) \subseteq L \subseteq \tilde B$
+such that $\lambda$ maps $L$ isomorphically onto all of $K$. Then $(L,B\cap L, \Sigma)$ is
+the desired lifting of $(K,A,S)$, that is, $(L,B\cap L,\Sigma) \subseteq (\tilde B, B, \Sigma)$
+and $\lambda$ restricts to an isomorphism $(L,B\cap L,\Sigma)\cong (K,A,S)$. We now shift our
+attention from $(B,\Sigma)$ (an expansion of the mixed characteristic valuation ring $B$)
+to $(\tilde B, L, B\cap L, \Sigma)$ which we view as the equal characteristic valuation ring
+$\tilde B$ equipped with a lifting of its expanded residue field $(K,A,S)$. Note that
+$B$ is definable in $(\tilde B, L, B\cap L, \Sigma)$ as follows: $B=\{x\in \tilde B: x-y\in \tilde \m
+\text{ for some } y\in B\cap L\}$.
+
+We now carry out the same construction with $(B',\Sigma')$, introducing
+$\tilde v'$, $\tilde \Gamma'$, $\tilde B'$, $K'$, $\k'$, $A'$, $S'$ and $L'$ in the
+same way we obtained the corresponding unaccented objects from $(B,\Sigma)$. As we indicated above it
+now suffices to show that $(\tilde B, L, B\cap L, \Sigma)\equiv(\tilde B', L', B'\cap L', \Sigma')$. Consider the rings $W(\k)$ and $W(\k')$ of Witt vectors over
+$\k$ and $\k'$, and for perfect subfields $F$ of $\k$ and $F'$ of $\k'$, consider the subrings
+$W(F)$ and $W(F')$ of $W(\k)$ and $W(\k')$, as well as the corresponding multiplicatively
+closed sets $S(F)\subseteq W(F)$ and $S(F')\subseteq W(F')$ that are mapped bijectively onto
+$F$ and $F'$ by the canonical maps $W(F) \to F$ and $W(F') \to F'$. In particular we have
+isomorphisms $(A,S) \cong (W(\k),S(\k))$ and $(A',S') \cong (W(\k'),S(\k'))$. Since
+$\k$ and $\k'$ are elementarily equivalent and $\aleph_1$-saturated, the isomorphisms
+$F\to F'$ between the countable $F\ee \k$ and $F'\ee \k'$ form a back-and-forth system
+between $\k$ and $\k'$. Each isomorphism $F\to F'$ of this system induces an
+isomorphism $(W(F),S(F))\to (W(F'),S(F'))$, thus giving rise to a back-and-forth system
+between $(W(\k),S(F))$ and $(W(\k'),S(\k'))$. Hence $(W(\k),S(F))\equiv (W(\k'),S(\k'))$, and so
+ $(A,S)\equiv(A',S')$. Therefore $(K,A,S)\equiv (K',A',S')$, and thus $(L,B\cap L, \Sigma)
+\equiv (L', B'\cap L', \Sigma')$. This allows us to apply Lemma 3 below to reach the desired
+conclusion $(\tilde B, L, B\cap L, \Sigma)\equiv(\tilde B', L', B'\cap L', \Sigma')$.
+This application also depends on the fact that $\Gamma \equiv \Gamma'$ implies $\tilde \Gamma \equiv
+\tilde \Gamma'$.\qed\end{proof}
+
+The lemma appealed to at the end is a variant of the well-known results of Ax, Kochen and Ersov, and
+can be proved in the same way, cf. \cite{ref1,ref3,ref4}. In this lemma
+the value group $\Gamma$ of a valuation ring $\Cal O$
+refers to the value group of the valuation $v$ on the fraction field of $\Cal O$ such that
+$v$ has $\Cal O$ as its valuation ring. This value group is considered as an ordered abelian group.
+
+
+\begin{lemma}Let $\Cal O$ and $\Cal O'$ be henselian valuation rings of
+equal characteristic
+$0$ with value groups $\Gamma$ and $\Gamma'$, and let $L \subseteq \Cal O$ and
+$L'\subseteq \Cal O'$ be fields that are mapped onto the
+residue fields of $\Cal O$ and $\Cal O'$ by the residue class maps
+$\Cal O \to \Cal O/\m(\Cal O)$
+and $\Cal O' \to \Cal O'/\m(\Cal O')$. Let $\Cal L$ be an extension of the
+language of rings, and
+let $L^*$ and ${L'}^*$ be expansions of the rings $L$ and $L'$ to $\Cal
+L$-structures. Then
+$$ (\Cal O, L^*) \equiv (\Cal O', {L'}^*) \Longleftrightarrow \Gamma
+\equiv \Gamma' \text{ and } L^*\equiv {L'}^* .$$
+\end{lemma}
+
+The following variant of the theorem can be obtained in the same way,
+by appealing to a corresponding variant of Lemma 3 (see \cite{ref4}). We
+let $\k$ and $\k'$ denote the residue fields of the valuation rings $B$ and $B'$,
+and let $\Gamma$ and $\Gamma'$ be their value groups as in the theorem.
+
+\begin{prop}Let $(B,\Sigma)$ and $(B',\Sigma')$ be models of $T$
+such that $(B,\Sigma)\subseteq (B',\Sigma')$ (so there are natural
+inclusions $\k \subseteq \k'$ and $\Gamma \subseteq \Gamma')$. Then
+$$(B,\Sigma)\preceq (B',\Sigma') \Longleftrightarrow \k\preceq \k'
+\text{ and }\Gamma\preceq \Gamma'.$$ \end{prop}
+
+\begin{rem}In Lemma 1 we described the $\Z$-linear
+relations among the elements of $S\subseteq A$. Another way to do this, in some respects more illuminating, is as follows.
+
+ First, any root of unity in $A$ belongs to $S$ and any
+tuple $\zeta = (\zeta_1,\dots,\zeta_n)$ ($n>0$) of roots of unity
+$\zeta_i\in A$ satisfies non-trivial $\Z$-linear relations. These relations produce
+ in certain obvious ways further relations, for example, for any $s\in S\setminus \{0\}$ the tuple $s\zeta$ satisfies the same $\Z$-linear relations
+as $\zeta$.
+
+ Secondly, an element $a\in A$ belongs to $S$ if and only if $F(a)=a^p$, where $F$ is the canonical lifting of the Frobenius map to
+an automorphism of $A$ (see \cite{ref5}).
+
+Using this last fact one can show,
+following \cite{ref2}, that all $\Z$-linear relations among elements of
+$S$ arise from the $\Z$-linear relations among the roots of unity in $A$.
+This was pointed out to me by Hrushovski.
+\end{rem}
+\begin{acknowledgement}The authors wish to thank H.\,Minkowski and
+D.\,Hilbert for stimulating discussions and encouragement.
+\end{acknowledgement}
+
+\begin{thebibliography}{[KuPr89]}
+\bibitem[A73]{ref1} Ax, J.:
+A metamathematical approach to some problems in number theory.
+AMS Symposium (1973) 161--190
+\bibitem[H]{ref2} Hrushovski, E.:
+The Manin-Mumford conjecture and the model theory of difference fields.
+Preprint
+\bibitem[Ko75]{ref3} Kochen, S.:
+The model theory of local fields. In: {\it Logic
+Conference, Kiel 1974} (Proceedings),
+Lecture Notes in Mathematics {\bf499}, Berlin 1975: Springer, pp.
+384--425
+\bibitem[KuPr89]{ref4} Kuhlmann, F.-V. and Prestel, A.:
+On places of algebraic function fields.
+J. reine angew. Math. {\bf400}, 185--202 (1989)
+\bibitem[S62]{ref5} Serre, J.-P.:
+{\it Corps Locaux}. Paris: Hermann, 1962
+\end{thebibliography}
+
+\end{document}