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\begin{document}
\title{A Method for Companionability,\\
Applied to Group Actions and Valuations%
\thanks{Joint work, initiated at the Nesin Mathematics Village,
with Ay\c se Berkman and with
\"Ozlem Beyarslan, Daniel Max Hoffmann, and G\"onen\c c Onay.
This is an edited version of the document submitted to,
and accepted by, the 11th Panhellenic Logic Symposium}}
\author{David Pierce}
\publishers{Mathematics Department\\
Mimar Sinan Fine Arts University\\
\url{dpierce@msgsu.edu.tr}\\
\url{http://mat.msgsu.edu.tr/~dpierce/}}
\maketitle
\begin{abstract}
A method for finding model companions
is applied to the theory of group actions
and to the theory of fields with both an automorphism and a valuation.
\end{abstract}
\section{The Method}
For every system of ordinary differential polynomial equations
over a differential field of characteristic $0$,
the consistency of the system---%
its solubility in some possibly larger differential field---%
is a first-order function of the parameters of the system.
Abraham Seidenberg \cite{MR0082487} showed this,
and from it,
Abraham Robinson \cite[\S5.5]{MR0153570}
derived
the theory $\dcf_0$ of \textbf{differentially closed fields}
of characteristic $0$,
which is to the theory $\df_0$ of all differential fields of characteristic $0$
as the theory $\acf$ of algebraically closed fields
is to the theory of all fields.
Specifically, $\dcf_0$ is the \emph{model completion} of $\df_0$.
To say what this means,
we denote by $\diag(\str M)$ the \textbf{diagram} of $\str M$,
namely the theory of structures in which $\str M$ embeds
\cite[\S2.1]{MR0153570};
this theory is axiomatized by the atomic and negated atomic sentences,
with parameters, that are true in $\str M$.
A theory $T^*$ is the \textbf{model completion} of a theory $T$
in the same signature \cite[\S4.3]{MR0153570} if
\begin{compactenum}[1)]
\item
$T\included T^*$,
\item
$T^*\cup\diag(\str M)$ is consistent
whenever $\str M\models T$,
\item
$T^*\cup\diag(\str M)$ axiomatizes a complete theory
whenever $\str M\models T$.
\end{compactenum}
When $T$ has the model completion $T^*$,
then immediately,
\begin{compactenum}[1)]
\item
every model of $T^*$ embeds in a model of $T$,
\item
every model of $T$ embeds in a model of $T^*$,
\item
$T^*$ is \textbf{model complete,}
that is, $T^*\cup\diag(\str M)$ is complete
whenever $\str M\models T^*$.
\end{compactenum}
Under these conditions alone,
$T^*$ is called the \textbf{model companion} of $T$
(the notion was introduced by ``Eli Bers'' in 1969
\cite[p.\ 609]{MR91c:03026}).
Given a theory $T$,
we define a \textbf{system} of $T$
to be a conjunction of atomic and negated atomic formulas in the signature of the theory.
$T$ is \textbf{inductive} if axiomatized by $\forall\exists$ sentences,
equivalently, every union of a chain of models is a model.
\begin{theorem}\label{thm:1}
If it exists,
the model-companion $\Trand\/^*$ of a theory $\Trand$
is axiomatized by $\Trand_{\forall}$ and sentences
\begin{equation*}
\Forall{\bm x}\Forall{\bm y}\Exists{\bm z}\bigl(\theta(\bm x,\bm y)\lto\phi(\bm x,\bm z)\bigr),
\end{equation*}
where
\begin{compactitem}
\item
$\phi$ is a system of atomic and negated atomic formulas,
\item
$\theta$ is from a set $\Theta_{\phi}$ of formulas, and
\item
for all models $\str M$ of $\Trand_{\forall}$
with parameters $\bm a$,
\begin{center}
$\theta(\bm a,\bm y)$ is soluble in $\str M$ for some $\theta$ in $\Theta_{\phi}$
$\iff$\\
$\phi(\bm a,\bm z)$ is soluble in a model of $\Trand_{\forall}\cup\diag{\str M}$.
\end{center}
\end{compactitem}
\end{theorem}
One can use Compactness to replace $\Theta_{\phi}$ with a single formula.
Also, if $\Trand$ has a model-\emph{completion,}
this single formula can be required to be quantifier-free
(and conversely):
this is what Robinson proved.
Seidenberg
had already shown that $\df$ met the condition on $\Trand$.
It was then observed, first by Blum \cite{MR0491149,MR99g:12006,MR2114160},
that not all systems of $\df$ need be considered.
Especially,
every ordinary differential polynomial equation can be written as
\begin{equation*}
f(\dots,\delta^jx_i,\dots)=0,
\end{equation*}
where $f$ is an ordinary polynomial;
and this equation
is equivalent to the result
of replacing each derivative with a new variable,
then conjoining the equation of the derivative with the variable:
\begin{equation*}
f(\dots,x_i^{(j)},\dots)=0
\land\bigwedge_{i,j}\delta x_i^{(j-1)}=x_i^{(j)}.
\end{equation*}
This approach of isolating the singulary operation $\delta$
is useful for other theories involving singulary operations,
specifically the theories of
\begin{compactenum}[1)]
\item
group actions (in work with Ay\c se Berkman),
\item
fields with automorphism and valuation
(in work with \"Ozlem Beyarslan, Daniel Max Hoffmann, G\"onen\c c Onay).
\end{compactenum}
The general result that we use is the following.
\begin{porism}
In the hypothesis of Theorem \ref{thm:1},
it is enough that $\phi(\vec x,\vec y)$ range over
a collection $\Phi$ of systems in the signature of $T$ containing,
\begin{compactenum}[(a)]
\item\myforall\
systems $\psi(\vec x,\vec u)$ of $T$,
\item\myforall\ models $\str M$ of $T$,
\item\myforall\ choices $\vec a$ of parameters from $M$,
\end{compactenum}
a system $\phi(\vec x,\vec u,\vec v)$
such that,
\begin{enumerate}[(i)]
\item
if $\Exists{\vec u}\psi(\vec a,\vec u)$
is consistent with $T\cup\diag{\str A}$,
then so is
$\Exists{\vec u}\Exists{\vec v}\phi(\vec a,\vec u,\vec v)$,
and
\item
$T\cup\diag(\str M)\proves
\Forall{\vec u}\Forall{\vec v}
\bigl(\phi(\vec a,\vec u,\vec v)\lto\psi(\vec a,\vec u)\bigr)$.
\end{enumerate}
\end{porism}
\section{Group Actions}
We can understand a \textbf{group action}
as kind of two-sorted structure $(\agp,\pset)$,
equipped with a function
\begin{equation*}
(\xi,y)\mapsto\myop{\xi}y
\end{equation*}
from $\agp\times\pset$ to $\pset$.
The structure should be a model of the theory $\GA$,
which has the following axioms:
\begin{gather*}
\Forall{\xi}\Exists{\eta}\Forall z(\myop{\xi}\myop{\eta} z=z\land\myop{\eta}\myop{\xi} z=z),\\
\Forall{\xi}\Forall{\eta}\Exists{\zeta}\Forall v
\myop{\xi}\myop{\eta} v=\myop{\zeta} v,\\
\Exists{\xi}\Forall y\myop{\xi} y=y.
\end{gather*}
In words, if the elements of $\agp$ are called \emph{functions,}
$\GA$ says
\begin{compactenum}[1)]
\item
every function has an inverse,
\item
any two functions have a composite,
\item
there is an identity function.
\end{compactenum}
There are no symbolized operations of actually taking inverses,
forming composites, or being the identity.
The theory $\FGA$ of \textbf{faithful} group actions
is axiomatized by $\GA$ along with
\begin{equation*}
\Forall{\xi}\Forall{\eta}\Exists z(\xi\neq\eta\lto\myop{\xi} z\neq\myop{\eta} z).
\end{equation*}
Of a theory $T$,
its \textbf{universal part} $T_{\forall}$
is the theory axiomatized by the universal sentences in $T$;
this is the theory of all substructures of models of $T$.
Then $T$ and $T_{\forall}$ have the same model companion,
if there is one.
To obtain a model companion for $\GA_{\forall}$,
it is enough to look at systems of equations $\myop{\xi}y=z$
and inequations $z\neq w$.
\begin{theorem}[Berkman, P.]\mbox{}
\begin{compactenum}
\item
$\GA$ and $\FGA$ are not inductive,
but they have the same universal part,
which is axiomatized by
\begin{equation*}
\Forall{\xi}\Forall y\Forall z(y\neq z\lto\myop{\xi} y\neq\myop{\xi} z),
\end{equation*}
meaning all functions are injective.
\item
Each of $\GA$ and $\FGA$ has a model companion, $\GA^*$,
which is axiomatized by $\GA_{\forall}$,
along with
\begin{gather*}
\Forall{\xi}\Forall y\Exists z\myop{\xi} z=y,\\
\Exists x\Exists yx\neq y,\\
\Forall{\bm x}\Exists{\bm{\xi}}
\left(\bigwedge_{i