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\title{Differentially closed fields}
\author{David Pierce}
\date{October 13, 2006}

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\fancyhead[RE]{[October 13,}
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\fancyhead[CE]{David Pierce}
\fancyhead[CO]{Differentially closed fields}
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[Added, June 6, 2007: I wrote these notes in preparation for the talk
  I gave at the MODNET meeting in Antalya.]

\begin{notation*}\mbox{}
  \begin{enumerate}
    \item
  $\str M$ is a structure; 
\item
$T$ is a first-order theory; 
\item
$\str M_M$ is $\str M$, with
all elements named.
  \end{enumerate}
\end{notation*}

\begin{definition}\mbox{}
  \begin{enumerate}
    \item
\fbox{$\diag{\str M}$} is the \emph{quantifier-free} theory of $\str M_M$.
    \item
\fbox{$T_{\forall}$} comprises the \emph{universal} consequences of $T$.
  \end{enumerate}
\end{definition}

\begin{lemma}\mbox{}
    \begin{enumerate}
\item
The class of structures in which $\str M_M$ embeds is elementary, and
its theory is $\diag{\str M}$.
      \item
    The class of substructures of models of $T$ is elementary, and its
theory is $T_{\forall}$.
    \end{enumerate}
  \end{lemma}

\begin{definition}\mbox{}
$T$ is \defn{model-complete} \cite{MR0472504} if $T\cup\diag\str M$ is
  complete  whenever $\str M\models T$. 
\end{definition}

\begin{theorem}[Robinson \cite{MR0472504}]
  The following are equivalent:
  \begin{enumerate}
    \item
$T$ is model-complete.
\item
$\included$ is $\elsub$ in $\Mod T$.
\item
$\included$ is $\elsub_1$ in $\Mod T$ (\defn{Robinson's test}).
  \end{enumerate}
\end{theorem}

\begin{minipage}{\textwidth}
\begin{definition}\mbox{}
  \begin{enumerate}
\item
    A model $\str M$ of $T$ is an \defn{existentially closed} model of
    $T$ if $\str M\elsub_1\str N$ whenever $\str M\included\str N$ and
    $\str N\models T$.
\item
  Suppose $T_{\forall}=T^*{}_{\forall}$.
\begin{enumerate}
\item
 $T^*$ is a \defn{model-completion} \cite{MR0153570} of $T$ if
$T\subseteq T^*$, and
  $T^*\cup\diag{\str M}$ is complete whenever $\str M\models T$.
\item
$T^*$ is a \defn{model-companion} \cite{MR0272613} of $T$ if $T^*$ is
  model-complete. 
  \end{enumerate}
  \end{enumerate}
\end{definition}
\end{minipage}
\mbox{}

Model-completions are model-companions.

\begin{definition}\mbox{}
  \begin{enumerate}
\item
    \fbox{$T_{\forall\exists}$} comprises the $\forall\exists$
    consequences of $T$.   
\item
$T$ is \defn{inductive} if the class of its models contains all unions
  of chains of its elements.
  \end{enumerate}
\end{definition}

\begin{lemma}[Chang--\L o\'s--Suszko \cite{MR0153570}]
$T$ is inductive if and only if $T_{\forall\exists}\vdash T$.
\end{lemma}

\begin{theorem}[Eklof \&\ Sabbagh \cite{MR0277372}]\mbox{}
  \begin{enumerate}
    \item
If $T$ is inductive, and the class of existentially closed models of
$T$ is elementary, then 
the theory of this class is the model-companion of $T$. 
\item
If $T$ has a model-companion, then its models are just the existentially
closed models of $T$.
  \end{enumerate}
\end{theorem}

\begin{example*}
  The theory of one-dimensional vector-spaces over algebraically
  closed fields is the model-completion of the theory of
  one-dimensional vector-spaces, and the model-companion of the theory
  of vector-spaces \cite{VS}.  (The signature is that of fields for
  the scalars,
  that of groups for the vectors, with a symbol for multiplication of
  vectors by scalars.)
\end{example*}

\begin{lemma}[Robinson]\mbox{}\label{lem:mc}
  \begin{enumerate}
    \item
Model-completions are unique.
\item
  Assume $T$ is inductive.  Then $T$ has a model-completion if and
  only if, for every open formula $\phi(\tuple x,\tuple y)$,
there is a formula
  $\widehat{\phi}(\tuple x,\tuple y)$ such that, for all models $\str
  M$ of $T$, for all $\tuple a$ from $M$,  
  \begin{equation*}
    \str M\models\Exists{\tuple x}\widehat{\phi}(\tuple x,\tuple a)
\Iff 
    T\cup\diag{\str M}\cup\{\Exists{\tuple x}\phi(\tuple x,\tuple a)\} 
   \text{ is consistent.} 
  \end{equation*}
If it exists, the model-completion is axiomatized by
\begin{equation*}
  T\cup\{\Forall{\tuple y} (\Exists{\tuple x}\widehat{\phi}(\tuple
  x,\tuple y)\to 
\Exists{\tuple x}\phi(\tuple x,\tuple y)):\phi(\tuple x,\tuple y)
  \text{ open}\}.  
\end{equation*}
  \end{enumerate}
\end{lemma}

\begin{definition}\mbox{}
  \begin{enumerate}
\item
$\ACF$ is the theory of algebraically closed fields.
\item
$\SCF$ is the theory of separably closed fields.
    \item
  $\DF$ is the theory of \defn{differential fields,} that is,
  structures $(K,D)$, where $K$ is a field, and $D$ is a
  \defn{derivation} of $K$, that is, an additive
  endomorphism obeying the Leibniz rule, $D(x\cdot y)=Dx\cdot y+x\cdot
  Dy$.  
\item
$\DPF$ is the theory of \defn{differentially perfect fields,} that is,
  models of $\DF$ satisfying $\Forall x\Exists y(p=0\land Dx=0\to
  y^p=x)$ for every prime $p$.
\item
If $T$ is one of these theories, then $T_p$ is the theory of models of
$T$ whose underlying fields have characteristic $p$.
  \end{enumerate}
\end{definition}

\begin{theorem}[Seidenberg \cite{MR0082487}]
  The function $\phi\mapsto\widehat{\phi}$ in Lemma~\ref{lem:mc}
  exists when $T$ is $\DF_p$. 
\end{theorem}

\begin{corollary}[Robinson]
  $\DF_0$ has a model-completion, $\DCF_0$.
\end{corollary}

\begin{theorem}[Wood \cite{MR48:8227}]
 If $p>0$, then $\DF_p$ has a model-companion, $\DCF_p$, which is the
 model-completion of $\DPF_p$.
\end{theorem}

\begin{lemma}[Blum \cite{MR0491149}]\mbox{}
  \begin{enumerate}
    \item
  Say $T\included T^*$, and $T_{\forall}\included T^*{}_{\forall}$.
  Then $T^*$ is the model-completion of $T$ if and only if
  \begin{equation*}
    \xymatrix{
\str M &\\
\str A \ar[u] \ar[r] & \str B \ar@{-->}[ul]_{\exists}
}
  \end{equation*}
when $\str A,\str B\models T$, and $\str M\models T^*$ and is $\size
B^+$-saturated. 
\item
If $T=T^*{}_{\forall}$, it is enough to assume that $\str B$ is generated
over $\str A$
by a single element (\defn{Blum's criterion}).
  \end{enumerate}
\end{lemma}

The second part holds by the analysis
\begin{equation*}
  \xymatrix{
\str M&&\\
\str A \ar[u] \ar[r] & \str A(a_0) \ar@{-->}[ul] \ar[r]
&\str A(a_0,a_1) \ar@{-->}[ull] \ar[r] & \dots \ar[r] 
& \str B \ar@{-->}@/_2ex/[ullll]}
\end{equation*}

\begin{theorem}[Blum \cite{MR0491149}]
The models of $\DCF_0$ are those models $(K,D)$ of $\DF_0$ that
contain solutions to every system
\begin{equation*}
  f(x,Dx,\dots,D^nx)=0\land g(x,Dx,\dots,D^{n-1}x)\neq0,
\end{equation*}
where $f(x_0,\dots,x_n)$ and $g(x_0,\dots,x_{n-1})$ are polynomials
over $K$, and $g\neq0$, and $f$ is non-constant in $x_n$.
\end{theorem}

\begin{theorem}[Wood \cite{MR50:9577}]
The models of $\DCF_p$ (where $p$ is prime) are those models $(K,D)$ of
$\DPF_p$ that contain solutions to every system
\begin{equation*}
  f(x,Dx,\dots,D^nx)=0\land g(x,Dx,\dots,D^{n-1}x)\neq0,
\end{equation*}
where $f(x_0,\dots,x_n)$ and $g(x_0,\dots,x_{n-1})$ are polynomials
over $K$, and $g\neq0$, and $f$ is non-constant in $x_n$ and separable.
\end{theorem}

In the proof, Blum's criterion is not used directly, all though in
principle it can be, if the operation for extracting $p$th roots
(where they exist) is given a symbol.

The case $n=0$ ensures that models of $\DCF_p$ are separably closed.
If $n>0$, then the condition on $f$ is that
$\partial_nf(x_0,\dots,x_n)\neq0$ has a solution.

Hence $\DF$ has a model-companion, axiomatized by:
\begin{enumerate}
\item\label{item:dpf}
$\DPF$,
  \item\label{item:scf}
$\SCF$,
\item\label{item:rest}
$\Forall{\tuple z}\Forall{\tuple y}\Exists
  x((\partial_n(y_0,\dots,y_n,\tuple z)\neq0\land
  g(y_0,\dots,y_{n-1},\tuple z)\neq0)\to(f(x,Dx,\dots,D^nx,\tuple
  z)=0\land g(x,Dx,\dots,D^{n-1}x,\tuple z)\neq0))$. 
\end{enumerate}

In characteristic $0$, \eqref{item:dpf} and \eqref{item:scf} become
$\DF_0$ and $\ACF_0$.  In this case, these theories can be retained,
while \eqref{item:rest} varies, and new model-complete theories of
differential fields can be obtained \cite{MR1990753}.

An alternative is to vary \eqref{item:scf}.

\begin{theorem}[Singer \cite{MR495120}]
  The theory of ordered differential fields (that is, ordered fields
  with a derivation) has a model-completion, whose models are those
  ordered differential fields $(K,<,D)$ that are real-closed, and
that  contain a solution to every system
  \begin{equation*}
    f(x,Dx,\dots,D^nx)=0\land \bigwedge_{j<m}g_j(x,Dx,\dots,D^nx)>0,
  \end{equation*}
provided that they contain a solution to
\begin{equation*}
f(x_0,\dots,x_n)=0\land\partial_nf(x_0,\dots,x_n)\neq0\land
\bigwedge_{j<m}g_j(x_0,\dots,x_n)>0.  
\end{equation*}
\end{theorem}

The following can be isolated from Singer's proof:

\begin{lemma}
  In the category of real-closed fields, if the system
  \begin{equation*}
    f(x_0,\dots,x_n)=0\land \partial_nf(x_0,\dots,x_n)\neq0
  \end{equation*}
has a solution at all, then it has a generic solution.
\end{lemma}

By Blum's criterion, to axiomatize the model-completion of a universal
theory, it is enough to understand solutions to systems in one
variable.  For differential fields, an alternative is the following.
For every differential field $(K,D)$, every primitive sentence
\begin{equation*}
  \Exists{\tuple x}\big(\bigwedge_ff(\tuple x,D\tuple
  x,\dots,D^n\tuple x)=0\land g(\tuple x,D\tuple
  x,\dots,D^n\tuple x)\big),
\end{equation*}
where $g$ and the $f$ are ordinary
polynomials over $K$, is equivalent, \emph{modulo}
$\DF\cup\diag{(K,D)}$, to
\begin{multline*}
  \Exists{(\tuple x_0,\dots,\tuple x_{n-1},\tuple y_0,\dots,\tuple
  y_{n-1})}
 \big(\bigwedge_ff(\tuple x_0,\dots,\tuple x_{n-1},\tuple
  y_{n-1})=0\land{}\\
{}\land g(\tuple x_0,\dots,\tuple x_{n-1},\tuple
  y_{n-1})\neq0\land
\bigwedge_{i<n-1}\tuple x_{i+1}=\tuple
  y_i\land\bigwedge_{i<n}D\tuple x_i=\tuple y_i\big). 
\end{multline*}
This sentence is a special case of the existentialization of the
formula
\begin{equation}\label{eqn:f}
  \big(\bigwedge_ff(\tuple x,\tuple
  y)=0\land g(\tuple x, \tuple y)\neq0\land D\tuple x=\tuple y\big).
\end{equation}
If this has a solution $(\tuple a,\tuple b)$ in some extension of
$(K,D)$, let that solution be a generic point of the variety $W$ over
$K$, and let $U$ be the open subset of this defined by $g(\tuple
x,\tuple y)\neq0$.  Let $V$ have generic point $\tuple a$.  Since
$D\tuple a=\tuple b$, we have that $(\tuple a,\tuple b)$ is in the
prolongation $\tau V$; so $U\included W\included \tau V$.  Any point
$(\tuple c,D\tuple c)$ of $U$ satisfies the formula~\eqref{eqn:f};
conversely, if $(\tuple c,\tuple d)$ is a generic point of $W$, then
$D$ extends to $K(\tuple c, \tuple d)$ so that $D\tuple c=\tuple d$.
Hence, `geometric' axioms for  $\DCF_0$ \cite{MR99g:12006}, for
$\DCF_p$ \cite{MR2119125}, and for $\CODF$ \cite{MR2173697}.


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\def\cprime{$'$}
\begin{thebibliography}{10}

\bibitem{MR0272613}
Jon Barwise and Abraham Robinson.
\newblock Completing theories by forcing.
\newblock {\em Ann. Math. Logic}, 2(2):119--142, 1970.

\bibitem{MR0491149}
Lenore Blum.
\newblock Differentially closed fields: a model-theoretic tour.
\newblock In {\em Contributions to algebra (collection of papers dedicated to
  Ellis Kolchin)}, pages 37--61. Academic Press, New York, 1977.

\bibitem{MR0277372}
Paul Eklof and Gabriel Sabbagh.
\newblock Model-completions and modules.
\newblock {\em Ann. Math. Logic}, 2(3):251--295, 1970/1971.

\bibitem{MR1990753}
E.~Hrushovski and M.~Itai.
\newblock On model complete differential fields.
\newblock {\em Trans. Amer. Math. Soc.}, 355(11):4267--4296 (electronic), 2003.

\bibitem{MR2119125}
Piotr Kowalski.
\newblock Derivations of the {F}robenius map.
\newblock {\em J. Symbolic Logic}, 70(1):99--110, 2005.

\bibitem{MR2173697}
Christian Michaux and C{\'e}dric Rivi{\`e}re.
\newblock Quelques remarques concernant la th\'eorie des corps ordonn\'es
  diff\'erentiellement clos.
\newblock {\em Bull. Belg. Math. Soc. Simon Stevin}, 12(3):341--348, 2005.

\bibitem{VS}
David Pierce.
\newblock Model-theory of vector-spaces over unspecified fields.
\newblock http://www.math.metu.edu.tr/\textasciitilde
  dpierce/papers/Vector-spaces/.

\bibitem{MR99g:12006}
David Pierce and Anand Pillay.
\newblock A note on the axioms for differentially closed fields of
  characteristic zero.
\newblock {\em J. Algebra}, 204(1):108--115, 1998.

\bibitem{MR0153570}
Abraham Robinson.
\newblock {\em Introduction to model theory and to the metamathematics of
  algebra}.
\newblock North-Holland Publishing Co., Amsterdam, 1963.

\bibitem{MR0472504}
Abraham Robinson.
\newblock {\em Complete theories}.
\newblock North-Holland Publishing Co., Amsterdam, second edition, 1977.
\newblock With a preface by H. J. Keisler, Studies in Logic and the Foundations
  of Mathematics, first published 1956.

\bibitem{MR0082487}
A.~Seidenberg.
\newblock An elimination theory for differential algebra.
\newblock {\em Univ. California Publ. Math. (N.S.)}, 3:31--65, 1956.

\bibitem{MR495120}
Michael~F. Singer.
\newblock The model theory of ordered differential fields.
\newblock {\em J. Symbolic Logic}, 43(1):82--91, 1978.

\bibitem{MR48:8227}
Carol Wood.
\newblock The model theory of differential fields of characteristic $p\not=0$.
\newblock {\em Proc. Amer. Math. Soc.}, 40:577--584, 1973.

\bibitem{MR50:9577}
Carol Wood.
\newblock Prime model extensions for differential fields of characteristic
  {$p\not=0$}.
\newblock {\em J. Symbolic Logic}, 39:469--477, 1974.

\end{thebibliography}


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