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\begin{document}

\title{Differential fields}
\author{David Pierce}
\date{2006}
\publishers{\url{http://mat.msgsu.edu.tr/~dpierce/}\\
\url{dpierce@msgsu.edu.tr}\\
\url{davut.deler@gmail.com}}
 \maketitle

\emph{This is a transcription, made in August, 2012 (last
  compiled \today), of handwritten
  notes that I used for a talk given at the mid-term Modnet meeting in
  Antalya in 2006.  The talk was given at the whiteboard, without
  slides.  I do not know how closely the talk followed these notes.
  Some sentences or paragraphs of my notes are bracketed in the
  manuscript, perhaps to indicate that I need not write them on the
  board.  I omit those brackets here.  Other parts of the notes are
  distinguished as being too much to talk about; those parts are
  omitted here.  The abstract of the talk was of
  course typed up and distributed at the time; it is displayed below.
  I have now made its defined terms bold, rather than italic.  It came
  with a bibliography, which is now printed as the bibliography of
  these notes.}  

\begin{abstract}
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 \textbf{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
\textbf{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 \textbf{model-companion}
%\cite{MR0277372} 
of $T$ if $T^*$ is model-complete.  A
\textbf{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 \textbf{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
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Various model-complete theories of differential fields are of ongoing
  interest.  It seems worthwhile to review them from the beginning.
  Some basic definitions are in the abstract.

\begin{example}
  The theory of the one-dimensional vector-spaces over algebraically
  closed fields is:
  \begin{itemize}
    \item
the \textbf{model-completion} of the theory of one-dimensional
vector-spaces,
\item
the \textbf{model-companion} of the theory of vector-spaces.
  \end{itemize}
\end{example}

A scalar-field can be made algebraically closed in only one way; but
  a vector-space of more than one dimension does not determine
  \emph{how} its dimensions can be collapsed when the scalar-field is
  enlarged. 

I shall talk about:
  \begin{itemize}
    \item
$\DF$, the theory of $(K,D)$, where $D\in\Der K$;
\item
$\DPF$, which is $\DF\cup\{\forall x\;\exists y\;(p=0\land Dx=0\lto
  x^p=y)\colon p\text{ prime}\}$.
  \end{itemize}

Another basic definition:

$\str M$ is an \textbf{existentially closed} model of $T$ if
\begin{equation*}
  \str M\included \str N\models T\implies\str M\preccurlyeq_1\str N,
\end{equation*}
that is, $\str M$ satisfies all quantifier-free formulas with
  parameters from $M$ that are satisfiable in $\str N$.

A characterization of model-companions:

\begin{theorem}[Eklof \&\ Sabbagh, 1971]
  If $T=T_{\forall\exists}$, then TFAE:
  \begin{itemize}
    \item
the class of existentially closed models of $T$ is elementary,
\item
$T$ has a model-companion,
\item
the model-companion of $T$ is the theory of the existentially closed
models of $T$.
  \end{itemize}
\end{theorem}

Model-completions meet a stronger condition:

\begin{theorem}[Robinson, $\leq$1963 \cite{MR0153570}]
  TFAE:
  \begin{itemize}
    \item
$T$ has a model-completion.
\item
$T=T_{\forall\exists}$, and there is $\phi\mapsto\hat{\phi}$ on
  existential (or just primitive) formulas such that, if $\str
  M\models T$ and $\vec a\in M^n$.
  \begin{equation*}
    \str M\models\hat{\phi}(\vec a)\iff\str M\included\str N\models
    T\cup\{\phi(\vec a)\} \text{ for some $\str N$},
  \end{equation*}
\item
the model-[completion] is
\begin{equation*}
  T\cup\{\forall\vec x\;(\hat{\phi}(\vec x)\lto\phi(\vec
  x))\colon\phi\text{ existential}\}.
\end{equation*}
  \end{itemize}
\end{theorem}

The immediate example is the theory of differential fields.
  Subscripts indicate characteristic; $\DF_0=\DPF_0$.

\begin{theorem}[Seidenberg]
  $\phi\mapsto\hat{\phi}$ as in Robinson's Theorem exists when $T$
  is $\DF_0$ or $\DPF_p$.
\end{theorem}

\begin{corollary}\mbox{}
  \begin{itemize}
    \item
\textnormal{Robinson, $\leq$1963 \cite{MR0153570}:}
  $\DF_0$ has a model-completion, $\DCF_0$.
\item
\textnormal{Wood, 1973 \cite{MR48:8227}:}
$\DPF_p$ has a model-completion, $\DCF_p$.
  \end{itemize}
\end{corollary}

For more comprehensible axioms, one can use:

\begin{theorem}[Blum, $\leq$1977 \cite{MR0491149}]
  TFAE:
  \begin{itemize}
    \item
$T^*$ is the model-completion of $T$,
\item
If $\str A,\str B\models T$; $\str M\models T^*$; $\str M$ is $\lvert
B\rvert^+$-saturated:
\begin{equation*}
  \xymatrix{
&\str M\\
\str A\ar[ur]\ar[r]&\str B\ar@{.>}[u]_{\exists}
}
\end{equation*}
  \end{itemize}
If, further, $T=T_{\forall}$, so substructures of models are models,
then the embedding of $\str A$ in $\str B$ can be analyzed as
\begin{equation*}
  \str A\to\str A(a_1)\to\str A(a_1,a_2)\to\dots\to\str B
\end{equation*}
where each structure is a model of $T$; so $\str B=\str A(a)$
suffices \emph{(Blum's Criterion)}.
\end{theorem}

Since $\DF_0$ is universal, Blum gets nice axioms for $\DCF_0$.  Wood
  gets similar axioms for $\DCF_p$, but \emph{cannot} use Blum's
  criterion, since $\DPF_p$ is not universal:

\begin{theorem}[Blum, $\leq$1977 \cite{MR0491149}, Wood, 1974 \cite{MR50:9577}]
  $(K,D)\models\DCF$ if and only if:
  \begin{itemize}
    \item
$(K,D)\models\DPF$,
\item
$K=K^{\operatorname{sep}}$,
\item
$(K,D)\models\exists x\;(f(x,Dx,\dots,D^{n+1}x)=0\land
  g(x,Dx,\dots,D^nx)\neq0)$ where $f$ and $g$ are ordinary
    polynomials over $K$, and $g\neq0$ and $\partial_{n+1}f\neq0$.
  \end{itemize}
\end{theorem}

Wood makes use of $r$, where
  \begin{equation*}
    \forall x;(r(x)^p=x\lor(Dx\neq0\land r(x)=0).
  \end{equation*}
Then $\DPF_p$ is universal, so Blum's Criterion can in principle be
used.  Rather, Wood uses a Primitive Element Theorem of Seidenberg.

Singer (1978 \cite{MR495120}) uses Blum's Criterion to get a
  model-completion of the theory of \emph{ordered} differential
  fields.  This means altering the condition
  $K=K^{\operatorname{sep}}$ (and then the last condition).
  Hrushovski and Itai (2003 \cite{MR1990753}) keep $K=K^{\operatorname{alg}}$ [\emph{sic}], but change the
  last condition to get many model-complete theories of differential
  fields.

An alternative approach:  First, we could have eliminated
  inequalities by the usual trick, $x\neq0\iff\exists y\;xy=1$.

Over $(K,D)$, a model of $\DPF$, TFAE:
\begin{gather*}
  \exists\vec x\;\bigwedge_ff(\vec x,D\vec x,\dots,D^n\vec x)=0,\\
\exists(\vec x_0,\dots,\vec x_n)\;(\bigwedge_ff(\vec x_0,\dots,\vec
x_n)=0\land\bigwedge_{i<n}D\vec x_i=\vec x_{i+1}).
\end{gather*}
The latter is an instance of
\begin{equation*}
  \exists(x_0,\dots,x_{n-1})\;(\bigwedge_ff(\vec
  x)=0\land\bigwedge_{i<k}Dx_i=g_i(\vec x)).
\end{equation*}
If this is witnessed by $\vec a$, WMA $(a_0,\dots,a_{k-1})$ is a
separating transcen\-dence basis of $K(\vec a)/K$.
\begin{equation*}
  \xymatrix{
\vec x\ar@{|->}[rr]^{\psi}\ar@{|->}[dd]_{\phi}&&(g_0(\vec x),\dots,g_{k-1}(\vec x))\\
&V(\vec a)\ar@{.>}[r]\ar@{->>}[d]^{\text{dominant, separable}}&\mathbb A^k\\
(x_0,\dots,x_{k-1})&\mathbb A^k&
}
\end{equation*}
$\DCF$ says: $V(\vec a)$ contains $P$ such that $D(\phi(P))=\psi(P)$
(P.\ \&\ Pillay 1998 \cite{MR99g:12006}). 

How do these ideas work in case of several derivations?
$\DF^m$ is the theory of $(K,\partial_0,\dots,\partial_{m-1})$, where
$\partial_i\in\Der K$ and $[\partial_i,\partial_j]=0$.

\begin{theorem}[McGrail, 2000 \cite{MR2001h:03066}]
  $\DF_0^m$ has a model-completion, $\DCF_0^m$.
\end{theorem}

\begin{proof}
  Use Blum's Criterion.  If $\sigma\in\upomega^m$, let
  $\partial^{\sigma}x$ denote
  \begin{equation*}
  \partial_0{}^{\sigma(0)}\dots\partial_{m-1}{}^{\sigma(m-1)}x.
  \end{equation*}
Let $\leq$ be the product-order on $\upomega^m$:
\begin{gather*}
  \sigma\leq\tau\iff\bigwedge_{i<m}\sigma(i)\leq\tau(i);\\
\lvert\sigma\rvert=\sum_{i<m}\sigma(i);\\
\begin{aligned}
\sigma\lessdot\tau\iff(\lvert\sigma\rvert,\sigma(0),\dots,\sigma(m-1))&<(\lvert\tau\rvert,\tau(0),\dots,\tau(m-1))\\
&\qquad\text{ lexicographically,}
\end{aligned}\\
K\langle a\rangle=K(\partial^{\sigma}a\colon\sigma\in\upomega^m)
\end{gather*}
If $\partial^{\sigma}a$ is algebraic over its $\lessdot$-predecessors,
then so is $\partial^{\sigma+\tau}a$; (and
$\partial^{\sigma+\tau}a\geq\partial^{\sigma}a$).  (Picture when
  $m=2$.)  [There was no picture in my notes.]

Hence $a$ is a generic zero of a system of finitely many equations.
That system can be chosen `coherent'; being coherent is first-order.
\end{proof}

How can we tell whether an arbitrary system has a solution?

\begin{example}
  $m=2$; does
  \begin{equation*}
    \partial^{(n,n)}x=x\land\partial^{(n-1,1)}x=\partial^{(0,n)}x
  \end{equation*}
have a solution?  $n=3$:
\begin{equation*}
  \xymatrix@!0{
a\ar@{-}[rrr]^{\stackrel{\partial_0}{\to}}\ar@{-}[ddd]_{\partial_1\downarrow}&&&b\ar@{-}[ddd]\\
&\cdot&\cdot&\\
&b&\cdot&\\
\ar@{-}[rrr]&&&a
}
\end{equation*}
Try differentiating to eliminate $\partial^{(n,n)}x$:
\begin{equation*}
  \xymatrix@!0{
a\ar@{-}[rrr]^{\stackrel{\partial_0}{\to}}\ar@{-}[ddd]_{\partial_1\downarrow}&&&b\ar@{-}[ddd]\ar@{-}[rrr]&&&\ar@{-}[ddd]\\
&\cdot&\cdot&&\cdot&a&\\
&b&\cdot&&\cdot&\cdot&\\
\ar@{-}[rrr]&&&a\ar@{-}[rrr]&&&
}
\end{equation*}
Check the common derivative of $b$ and $a$:
\begin{equation*}
  \xymatrix@!0{
a\ar@{-}[rrr]^{\stackrel{\partial_0}{\to}}&&&b\ar@{-}[ddd]\ar@{-}[rrr]&&&\ar@{-}[ddd]&c\\
c\ar@{-}[dd]&\cdot&\cdot&&\cdot&a&&\cdot\\
&b&\cdot&&\cdot&c&&\cdot\\
\ar@{-}[rrr]&&&a\ar@{-}[rrr]&&&&\cdot
}
\end{equation*}
Check the common derivative of $a$ and $c$:
\begin{equation*}
  \xymatrix@!0{
a\ar@{-}[rr]&&d&b\ar@{-}[ddd]\ar@{-}[rrr]&&&\ar@{-}[ddd]&c\\
c&\cdot&\cdot&&\cdot&a&&d\\
d&b&\cdot&&\cdot&c&&\cdot\\
\ar@{-}[rrr]&&&a\ar@{-}[rrr]&&&&\cdot
}
\end{equation*}
A new condition is imposed; what we started with cannot be a solution.
\end{example}

\begin{theorem}
  For every $m$ and $n$, there is $M$ such that, for all models
  $(K,\partial_0,\dots,\partial_{m-1})$ of $\DF_0^m$, for all fields
  $K(a^{\sigma}\colon\sigma\leq(Mn,\dots,Mn))$, if the $\partial_i$
  extend so that
  \begin{align*}
    \partial_ia^{\sigma}&=a^{\sigma+\bm i},&(\bm i(j)&=\updelta_{i\;j}),
  \end{align*}
then
$(K,\partial_0,\dots,\partial_{m-1})\included(L,\tilde{\partial}_0,\dots,\tilde{\partial}_{m-1})\models\DF_0^m$,
where
\begin{equation*}
  K(a^{\sigma}\colon\sigma\leq(n,\dots,n))\included L
\end{equation*}
and $\tilde{\partial}_ia^{\sigma}=a^{\sigma+\bm i}$.
\end{theorem}

Here $M\sim m^{m^{\ddots^m}}$ (a stack of $n$ exponents); but I have
some hope that $M$ can be $m$.

See earlier example.

Differential forms\dots
\relscale{0.8}
\begin{thebibliography}{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{thebibliography}

\end{document}