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\begin{document}  

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\titleoftalk{Differential fields}

\speaker{David Pierce} 

In a differential field, how can we tell whether all consistent
systems of equations and inequations have solutions?  I shall review
the history of answers to this question, and I shall
update the accounts in \cite{MR2000487,MR2114160}.

To begin with the Robinsonian beginnings, I remind or
inform the reader-listener of the following.
The class of substructures of models of a theory $T$ is elementary,
and its theory is $T_{\forall}$.  The class of structures in which a
structure $\mathfrak M$ embeds is elementary, and its theory is
$\operatorname{diag}(\mathfrak M)$.  The class of models of $T$ is
closed under unions of chains if and only if $T=T_{\forall\exists}$
\cite[3.4.7]{MR0153570}.
The theory $T$ is called \emph{model-complete} \cite{MR0472504} if
$T\cup\operatorname{diag}(\mathfrak M)$ is complete whenever
$\mathfrak M\models T$.  If $T\subseteq T^*$, and
$T_{\forall}=T^*{}_{\forall}$, then $T^*$ is the
\emph{model-completion} \cite{MR0153570} of $T$ if
$T^*\cup\operatorname{diag}(\mathfrak M)$ is complete whenever 
$\mathfrak M\models T$; but $T^*$ is merely the \emph{model-companion}
%\cite{MR0277372} 
of $T$ if $T^*$ is model-complete.  A
\emph{derivation} of a field $K$ is an additive endormorphism $D$ of
$K$ that respects the Leibniz rule, $D(x\cdot y)=Dx\cdot y+x\cdot
Dy$.  A \emph{differential field} is a field equipped with one or more
derivations.

\nocite{MR0491149} % Blum
\nocite{MR48:8227} % Wood: model theory of diff. fields
\nocite{MR50:9577} % Wood: prime model extensions
\nocite{MR495120}  % Singer
\nocite{MR99g:12006} % PP
\nocite{MR1990753} % HI
\nocite{MR2173697} % MR
\nocite{GR}        % GR
\nocite{MR2001h:03066} % McGrail
\nocite{MR1807840} % Yaffe
\nocite{MR2159694} % Tressl
\nocite{MR2119125} % Kowalski

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\begin{ourbiblio}{999}

\bibitem[R1]{MR0472504}
Abraham Robinson.
\newblock {\em Complete theories}.  1956.

\bibitem[R2]{MR0153570}
Abraham Robinson.
\newblock {\em Introduction to model theory.}
% and to the metamathematics of algebra}.
1963.

\bibitem[W1]{MR48:8227}
Carol Wood.
\newblock The model theory of diff.\ fields of characteristic $p\not=0$.
1973.

\bibitem[W2]{MR50:9577}
Carol Wood.
\newblock Prime model extensions for diff.\ fields of char.\ {$p\not=0$}.
1974.

\bibitem[B]{MR0491149}
Lenore Blum.
\newblock Differentially closed fields: a model-theoretic tour.
1977.

\bibitem[S]{MR495120}
Michael~F. Singer.
\newblock The model theory of ordered differential fields.
1978.

\bibitem[PP]{MR99g:12006}
D. Pierce and Anand Pillay.
\newblock A note on the axioms for differentially closed fields of
  characteristic zero.
1998.

\bibitem[McG]{MR2001h:03066}
Tracey McGrail.
\newblock The model theory of differential fields with finitely many commuting
  derivations.
2000.

\bibitem[Y]{MR1807840}
Yoav Yaffe.
\newblock Model completion of {L}ie differential fields.
2001.

\bibitem[HI]{MR1990753}
E.~Hrushovski and M.~Itai.
\newblock On model complete differential fields.
2003.

\bibitem[P1]{MR2000487}
D. Pierce.
\newblock Diff.\ forms in the model theory of diff.\ fields.
2003.

\bibitem[P2]{MR2114160}
D. Pierce.
\newblock Geometric characterizations of e.c.\ fields with
  operators.
2004.

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

\bibitem[T]{MR2159694}
Marcus Tressl.
\newblock The uniform companion for large diff.\ fields of char.\ 0.
2005.

\bibitem[MR]{MR2173697}
Christian Michaux and C{\'e}dric Rivi{\`e}re.
\newblock Quelques remarques concernant la th\'eorie des corps ordonn\'es
  diff\'erentiellement clos.
2005.

\bibitem[GR]{GR}
Nicolas Guzy and C{\'{e}}dric Rivi{\`{e}}re.
\newblock Principle of differential lifting for theories of differential fields
  and Pierce--Pillay axiomatization.
20xx.

\end{ourbiblio}

\speakeraddress{Orta Do\u gu Teknik \"Universitesi, Ankara}

\email{dpierce@metu.edu.tr}

\web{www.math.metu.edu.tr/~dpierce}  


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