%% The 43re Iranian Mathematics Conferencs, University of Tabriz, Tabriz, Iran. %% \documentclass[12pt]{imc43} \usepackage{stmaryrd} % for \trianglelefteqslant %%%%%%%%%%%%%%%%%TITLE%%%%%%%%%%%%%%%%%%% \title[Differential fields]{Model-theory of differential fields} \author{David Pierce \university{Mimar Sinan Fine Arts University} } \begin{document} \maketitle \begin{abstract} After some generalities about model-theory, I give a specific result about differential fields: for every natural number, the theory of fields with that number of commuting derivations has a model-companion. This is so because, if a system of partial differential equations is given, there is a way to tell in finitely many steps whether the system is soluble, and moreover the number of steps is independent of the parameters of the system. \keyword{model-theory, differential field, model-completeness, model-companion} \subject{03C10} \end{abstract} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Model-theory} I consider model-theory to be the study of \emph{structures} as \emph{models} of \emph{theories.} This definition has three terms that need explanation. For the moment, a \emph{theory} is just a set of sentences of a formal logic. If each of those sentences is true in a structure, that structure is a \emph{model} of the theory. The logic is usually first-order, and will always be so here; this means variables stand for individuals, never sets as such, and moreover sentences are finite: only finitely many variables appear in a given sentence, and all conjunctions and disjunctions are finitary. A \emph{structure} then consists of one or more sets, together with various relations and operations on those sets, along with distinguished individual elements of those sets. None of these additional features is actually required to be present: a bare set is a structure. But groups, rings, ordered fields, and vector-spaces are also structures. One non-example is a topological space, considered as a set $\Omega$ with the closure operation $X\mapsto\overline X$: the problem here is that $X$ ranges, not over elements of $\Omega$, but over sets of elements of $\Omega$. However, certain topological spaces are essential to model theory: first-order formulas in a given number of free variables determine a Boolean algebra, called a Lindenbaum algebra, and the elements of the Stone space of this algebra are called types. One reason to use first-order logic is that it has a compactness theorem: if every finite subset of a theory has a model, then so does the whole theory. One might see this as a restriction: it implies for example that there is no theory whose models are precisely the torsion groups. Indeed, suppose $T$ is a theory of which every torsion group is a model. We introduce a new constant $g$ and, for each positive integer $n$, a sentence $\sigma_n$, namely $g^n\neq1$. Every finite subset of $T\cup\{\sigma_k\colon k\in\mathbb N\}$ has a model, namely $\mathbb Z/n\mathbb Z$ for sufficiently large $n$; therefore the whole set has a model, and in this model, $g$ must be interpreted as a non-torsion element. So compactness can be used to show some limitations of first-order logic. But it can also be used to make rigorous the intuitive approach of Newton and Leibniz to calculus \cite{MR1373196}. A structure has a \emph{signature,} namely a set of symbols for the distinguished relations, operations, and individuals of the structure. Then a sentence $\sigma$ can be true or false in a structure $\mathfrak M$ only if the non-logical symbols in $\sigma$ come from the signature of $\mathfrak M$. A set $\Gamma$ of sentences \emph{entails} a sentence $\sigma$, and $\sigma$ is a \emph{logical consequence} of $\Gamma$, if $\sigma$ is true in every model of $\Gamma$. Now we can say that $\Gamma$ is a \emph{theory} if (and only if) it contains all of its logical consequences. In any case, if $T$ is the set of logical consequences of $\Gamma$, then $\Gamma$ is a set of \emph{axioms} for $T$. The theory $T$ is then \emph{complete} if, for every sentence $\sigma$ of its signature, $T$ entails either $\sigma$ or its negation. Every structure has a theory, namely the set of sentences that are true in the structure. This theory is automatically complete. However, by G\"odel's Incompleteness Theorem of 1931, there is no method for writing down a set of axioms for the theory of $\mathbb N$ in the signature $\{+,\times\}$. Nonetheless, by slightly earlier work of Tarski's student Presburger, there \emph{is} such a method in the smaller signature $\{+\}$. In a word, the structure $(\mathbb N,+)$ is \emph{`tame'}, but $(\mathbb N,+,\times)$ is not tame. An early theme of model-theory is just the identification of tame structures \cite{MR0472504}. Further examples are $(\mathbb R,+,\times,\leqslant)$, $(\mathbb C,+,\times)$, and $(\mathbb Q,\leqslant)$. By the compactness theorem, every theory with infinite models has infinitely many non-isomorphic models. The theory $\mathrm{ACF}_0$ of $(\mathbb C,+,\times)$ has countably many non-isomorphic countable models, but just one model of each uncountable cardinality. The theory $\mathrm{LO}^*$ of $(\mathbb Q,\leqslant)$ has one countable model, but $2^{\kappa}$ models in each uncountable cardinality $\kappa$. An ongoing task of model-theory is to understand the combinatorial properties that effect distinctions such as the one just described between the class of models of $\mathrm{ACF}_0$ and of $\mathrm{LO}^*$. For example, $\mathrm{LO}^*$ is \emph{unstable,} because of the ordering; no such ordering can be defined in $\mathrm{ACF}_0$, so this theory is \emph{stable}; in fact it is $\omega$-stable, because its Stone spaces in countably many parameters are themselves countable. Abraham Robinson investigated a variant of completeness that he called \emph{model completeness.} To define this, we first define the \emph{diagram} of a structure: this is the set of \emph{quantifier-free} sentences, with parameters from the structure, that are true in the structure. For example, the diagram of $(\mathbb N,+,\times)$ is generated by the usual addition and multiplication tables learned at school. Then a theory $T$ is \emph{model-complete} if the set $T\cup\operatorname{diag}(\mathfrak M)$ axiomatizes a complete theory whenever $\mathfrak M$ is a model of $T$. For example, the theory $\mathrm{ACF}$ of algebraically closed fields is model-complete: this follows because for every model $K$, the theory (axiomatized by) $\mathrm{ACF}\cup\operatorname{diag}(K)$ has only one model in each cardinality greater than that of $K$ itself. The theory $\mathrm{ACF}$ is not itself complete, but it becomes complete when an axiom specifying the characteristic of a model is added. Indeed, the new theory is complete because it is model-complete and it has a model that embeds in all other models. Because then $\mathrm{ACF}\cup\operatorname{diag}(K)$ is complete for every field $K$, the theory $\mathrm{ACF}$ is called the \emph{model-completion} of the theory of fields. A slightly more general notion is that of \emph{model-companion}: a theory $T^*$ is a model-companion of $T$ if every model of the one theory embeds in a model of the other, and moreover $T^*$ is model-complete. If $\mathrm{LO}$ is the theory of linear orders, then $\mathrm{LO}^*$ as defined above is indeed its model-companion. The theory of groups has no model-companion; neither does the theory of fields with a distinguished algebraically closed subfield. Suppose $T$ is a theory such that the union of an increasing chain of models is itself a model; equivalently, like most structures studied in algebra, $T$ is axiomatized by $\forall\exists$ sentences. The \emph{existentially closed} models of $T$ are those models $\mathfrak M$ such that, for every quantifier-free formula $\phi$ in the signature of $\mathfrak M$ with parameters, if $\phi$ has a solution in some extension of $\mathfrak M$ that is a model of $T$, then $\phi$ already has a solution in $\mathfrak M$ itself. If (and only if) there is a theory whose models are precisely the existentially closed models of $T$, then this theory is the model-companion of $T$ \cite{MR0277372}. The theory of fields with a distinguished algebraically closed subfield has no model-companion, because the existentially closed models of this theory are the algebraically closed fields \emph{of transcendence-degree one} over an algebraically closed subfield, and transcendence-degree can be given no first-order characterization. \section{Differential fields} High-school algebra and calculus combine in \emph{differential fields:} these fields with one or more \emph{derivations,} namely operations $D$ with the algebraic properties of `taking the derivative': $D(x+y)=Dx+Dy$, and $D(x\cdot y)=Dx\cdot y+x\cdot Dy$. Let $\mathrm{DF}$ be the theory of fields with a single derivation. A required characteristic can be indicated by a subscript. Using an elimination result of Seidenberg, Robinson found a model-companion of $\mathrm{DF}_0$, but its axioms were not illuminating. The model-companion of field-theory needs only axioms saying that every non-constant polynomial \emph{in one variable} has a root. Lenore Blum showed that a similar result was possible for $\mathrm{DF}_0$. Meanwhile Carol Wood found a model-companion for $\mathrm{DF}_p$ when $p$ is positive. Combining these results yields the following: \begin{theorem}[Robinson, Blum, Wood] A model $(K,D)$ of $\mathrm{DF}$ is existentially closed if and only if each of the following conditions holds. \begin{enumerate} \item $K$ is separably closed. \item $(K,D)$ is \emph{differentially perfect} (in positive characteristic $p$, if $Dx=0$, then $x$ has a $p$th root). \item The sentence \begin{equation*} \exists x\;\bigl(f(x,Dx,\dots,D^{n+1}x)=0\land g(x,Dx,\dots,D^nx)\neq0\bigr) \end{equation*} is true in $(K,D)$ whenever $f$ and $g$ are ordinary polynomials over $K$, in tuples $(x^0,\dots, x^{n+1})$ and $(x^0,\dots,x^n)$ of variables respectively, such that \begin{align*} \frac{\partial f}{\partial x^{n+1}}&\neq0,& g&\neq0. \end{align*} \end{enumerate} Hence $\mathrm{DF}$ has a model-companion. \end{theorem} Following Robinson, we may call this model-companion $\mathrm{DCF}$. Its \emph{completions}---the complete theories that include it---are obtained by specifying a characteristic. The completion $\mathrm{DCF}_0$ is $\omega$-stable; when $p$ is positive, $\mathrm{DCF}_p$ is not $\omega$-stable, but is stable \cite{MR0344116}. An alternative way to simplify the axioms of $\mathrm{DCF}$ is to consider, not systems in one variable, but first-order systems---first-order, not in the sense of logic, but in the sense of having only single applications of the derivation. Then the models of $\mathrm{DCF}$ can be described geometrically \cite{MR99g:12006,MR2114160}: \begin{theorem} A differential field $(K,D)$ is existentially closed if and only if each of the following conditions holds. \begin{enumerate} \item $K$ is separably closed. \item $(K,D)$ is differentially perfect. \item\label{item:rat-pt} For every variety $V$ over $K$, if there are rational maps $\phi$ and $\psi$ from $V$ to $\mathbb A^n$ for some $n$, where $\phi$ is dominant and separable, then $V$ has a $K$-rational point $P$ such that $\phi$ and $\psi$ are regular at $P$, and $D\circ\phi(P)=\psi(P)$. \end{enumerate} In Condition~\ref{item:rat-pt}, it is sufficient to assume $n=\dim V$. \end{theorem} \section{Several derivations} Given a positive integer $m$, we may let $\text{$m$-DF}$ be the theory of fields with $m$ commuting derivations. So in this theory, equations are so-called \emph{partial differential equations.} As usual, a required characteristic can be given by a subscript. The existence of a model-companion of $\text{$m$-DF}_0$---call it $\text{$m$-DCF}_0$---was established by Tracey McGrail; alternative characterizations and generalizations (still in characteristic $0$) were given by Yaffe and by Tressl. However, as with Robinson's original account of $\mathrm{DCF}_0$, none of these descriptions of $\text{$m$-DCF}_0$ is perspicuous. It appears that neither of the methods described above for simplifying the axioms of $\mathrm{DCF}$ is useful for $\text{$m$-DCF}$. Nonetheless, we have the theorem below (in which no characteristic is specified) \cite{2007arXiv0708.2769P}. Notation is as follows. \begin{itemize} \item $\omega$ is the set of von-Neumann natural numbers, so that an element $n$ is the set $\{0,\dots,n-1\}$. \item If $\xi\in\omega^m$, that is $\xi=(\xi(0),\dots,\xi(m-1))$, then \begin{align*} \lvert\xi\rvert&=\sum_{i