\documentclass{beamer}

\mode<presentation> {
  \usetheme{Montpellier}
  \setbeamercovered{transparent}
}

\usepackage[english]{babel}
\usepackage[utf8x]{inputenc}
\usepackage{times}
\usepackage{color}
\usepackage[T1]{fontenc}
\newcommand{\del}{\partial}
\newcommand{\lra}{\longrightarrow}
\newcommand{\sra}[1]{\stackrel{#1}{\lra}}
\newcommand{\Cl}{\mathcal{C}\ell}
\newcommand{\N}{\mathcal{N}}
\renewcommand{\P}{\mathcal{P}}
\renewcommand{\phi}{\varphi}
\renewcommand{\O}{\mathcal{O}}
\newcommand{\p}{\mathfrak{p}}
\newcommand{\f}{\mathfrak{f}}
\renewcommand{\S}{\mathfrak{S}}
\newcommand{\Ot}{O\tilde{~}}
\newcommand{\Et}{\tilde{E}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\FB}{\overline{\mathbb{F}}}
\newcommand{\FF}{\mathbb{F}}
\renewcommand{\d}{\mathrm{d}}
\newcommand{\End}{\mathrm{End}}
\newcommand{\Gal}{\mathrm{Gal}}
\newcommand{\norm}{\mbox{Norme}}
\newcommand{\atanh}{\mbox{atanh}}
\newcommand{\trace}{\mbox{Trace}}
\newcommand{\frob}[2]{\left(\frac{#1}{#2}\right)}
\newcommand{\val}{v}
\newcommand{\Disc}{\mathrm{Disc}}
\newcommand{\spec}{\mathrm{Spec}}
\newcommand{\mat}[4]{\left(\begin{array}{cc} #1 & #2 \\ #3 & #4 \end{array}\right)}
%\renewcommand{\theorem}{}
\newtheorem{theorema}{Theorem}
\newtheorem{prop}{Proposition}
\newtheorem{lemmaa}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{princip}[theorem]{Principle}
\newtheorem{algorithm}[theorem]{Algorithm}
\renewcommand{\theorem}{\theorema}
\renewcommand{\lemma}{\lemmaa}
\def\e{*}
\catcode`*=13
\def*{\*}

\title{Symbolic integration in finite chararcteristic}

\author{B.~Allombert}
\institute
{
  IMB \\
  CNRS/Université Bordeaux 1
}

\date{2014/12/2}

\subject{Exposé}
%\logo{\pgfimage[height=1cm]{um2.png}}
%\logo{\pgfuseimage{um2}}
%\logo{\includegraphics[height=1.5cm]{um2.png}}
\AtBeginSubsection[] {
  \begin{frame}<beamer>{Lignes directrices}
    \tableofcontents[currentsection,currentsubsection]
  \end{frame}
}

%\beamerdefaultoverlayspecification{<+->}

\begin{document}
\begin{frame}
  \titlepage
\end{frame}

\begin{frame}{Lignes directrices}
  \tableofcontents
  % Vous pouvez, si vous le souhaiter ajouter l'option [pausesections]
\end{frame}

\section{Introduction}

\begin{frame}{Elementary functions}
Traditionnaly, a function from $\CC$ to $\CC$ is said to be \emph{elementary}
if it can be written as a composite of multivariate algebraic functions,
the exponential and the logarithm.

\par

Liouville in 1838 showed that some elementary functions do not admit an
elementary antiderivative. Two important examples are $x\mapsto \exp(x)/x$
and $x\mapsto \sqrt{x^4+1}$. Risch in 1968 gives an algorithm to decide if
an elementary function admit an elementary derivative.
\end{frame}

\begin{frame}{Derivations}

Ritt in 1930 gives an algebraic definition of elementary functions in
term of differential fields which do not require functions to be composable,
or even to be have values at points.

Let $K$ be a field. A \emph{derivation} on $K$ is a group homomorphism
$\del=\del_K$ from $K$ to $K$ such that for all $a$, $b$ in $K$ the
following holds:
$$\del(a*b) = \del(a)*b+a*\del(b)\enspace.$$

Example: if $K$ is a field, then $f\mapsto f'$ is a derivation over $K(X)$.

\end{frame}

\section{Elementary extensions}

\begin{frame}{Differential fields}

A couple $(K,\del_K)$ is a \emph{differential field} if $K$ is a field and
$\del_K$ is a derivation on $K$.

A \emph{differential extension} of $(K,\del_K)$ is a differential field 
$(L,\del_L)$ such that $L/K$ is a field extension and $\del_L|_K=\del_K$.

The kernel of $\del_K$ is a field called the field of constants of
$(K,\del_K)$, and is denoted by $C(K)$. 

\end{frame}

\begin{frame}{Elementary extensions}

\begin{theorem}
If $(K,\del_K)$ is a differential field and $L/K$ is an algebraic separable
extension, then $\del_K$ can be extended in a unique way to a derivation
$\del_L$ on $L$ so that $(L,\del_L)$ is a differential extension of $(K,\del_K)$.
\end{theorem}

\end{frame}

\begin{frame}{Elementary extensions}

\begin{definition}
A differential extension $(L,\del_L)$ of $(K,\del_K)$ is logarithmic
of type ``$log(u)$'' if there exists $u\in K^{\*}$ and $y\in L$ generating $L$
over $K$ such that $\del(u)=\del(y)*u$.
\end{definition}

\begin{definition}
A differential extension $(L,\del_L)$  of $(K,\del_K)$ is exponential
of type ``$exp(u)$'' if there exists $u\in K^{\*}$ an $y\in L$ generating $L$ over $K$
such that $\del(y)=\del(u)*y$.
\end{definition}

\end{frame}

\begin{frame}{Elementary extensions}

\begin{definition}
A differential extension $(L,\del_L)$ of $(K,\del_K)$ is elementary if there 
exists a tower of differential extensions 
$L_0=K\subseteq L_1 \ldots\subseteq L_n = L$ such that
each extension $L_{i+1}/L_i$ is either algebraic separable, logarithmic,
or exponential.
\end{definition}

Traditional elementary functions are just elements of elementary extensions
of $\CC(X)$.

\end{frame}

\section{Differential fields in finite characteristic}

\begin{frame}{Differential fields in finite characteristic}
Let $(K,\del_K)$ be a differential field of characteristic $p$, then
\begin{itemize}
\item for all $a\in K$, $\del(a^p)=0$ (i.e. $K^p\subseteq C(K)$).
\item for all positive integers $e$, $\del^{p^e}$ is a derivation.
\end{itemize}
\begin{definition}[$p$-polynomials]
A polynomial $P$ over a field $K$ is a $p$-polynomial if it can be written as 
$P=\sum_{j=0}^n a_j*X^{p^j}$ with $a_j\in K$ for $0\leq j\leq n$.
\end{definition}
If $P$ is a $p$-polynomial, then $P(\del_K)$ is a derivation on $K$. 
\end{frame}

\begin{frame}{Symbolic integration in finite characteristic}
The goal of this talk is to prove 
\begin{theorem}
Let $k$ be a field of characteristic $p$,  $(K,\del_K)$ an elementary
extension of $(k(X),F->F')$, $f$ an element of $K$, then there exists a
differential extension of logarithmic type $(L,\del_L)$ of $(K,\del_K)$
and $g\in L$ such that $\del_L(g)=f$.
\end{theorem}
\end{frame}

\section{Linear fields}

\begin{frame}{Linear differential equations}

\begin{definition}
Let $(K,\del_K)$ be a differential field, let $y$ be an element of $K$, and
let $C$ be a sub-differential field of $K$.
We will say that $y\in K$ satisfies a linear differential equation with 
coefficients in $C$ if there exists a non-zero polynomial $P\in C[X]$ such 
that $P(\del)(y)=0$.
\end{definition}

\begin{definition}
A differential field $(K,\del_K)$ is \emph{linear} if every element satisfies a
linear differential equation with coefficients in $C(K)$.
\end{definition}

\end{frame}

\begin{frame}{$p$-polynomials}

\begin{lemma}
If $K$ is a field and $I$ a non-zero ideal of $K[X]$, then $I$ contains a
non zero $p$-polynomial.
\end{lemma}
(follow from $K[X]/I$ being finite dimensionnal).

\begin{lemma}\label{peqn}
Let $(K,\del_K)$ be a differential field, and $(y_i)_{i=1}^n$ be a family of
elements of $K$. If the $y_i$ satisfy a linear differential equation with 
constant coefficients for all $1\leq i\leq n$, then there exists a non-zero 
$p$-polynomial $P$ in $C(K)[X]$ such that $P(\del)(y_i)=0$ for all 
$1\leq i\leq n$.
\end{lemma}
\end{frame}

\begin{frame}{$p$-polynomials}
\begin{prop}
Let $(K,\del_K)$ be a differential field.
The set $\mbox{Lin}(K)$ of elements of $K$ that satisfy a
linear differential equation with coefficients in $C(K)$
is a sub-differential field of $K$.
\end{prop}
This follows from the fact that $C(P(\del))$ is a field if $P$ is a
$p$-polynomial.
\end{frame}

\begin{frame}{Elementary extensions are linear}

\begin{theorem}\label{diffeqn}
Let $(K,\del_K)$ be a linear differential field, and let $(L,\del_L)$ be a
differential extension of $(K,\del_K)$. If $L/K$ is elementary, 
then $(L,\del_L)$ is linear.
\end{theorem}

\begin{lemma}
The derivatives of the classical differential equation $y'=u'*y$
are given by the multivariate Bell polynomials.
In characteristic $p$, Carlitz~Formula~(1.4) gives the equation
\begin{equation}
\del^{p^r}(y)=(\sum_{i=0}^r {\del^{p^i}(u)}^{p^{r-i}})*y\enspace.
\end{equation}
\end{lemma}

\end{frame}

\begin{frame}

\begin{example}
Let $K=(\FF_p(E),\del_K)$ be such that $\del_K(E)=E$,
and let $L$ be the differential extension $(\FF_p(E,F),\del_L)$ where
$\del_L(F)=E*F$. We find that $\del_L^p(F)-\del_L(F)=E^p*F$.
\end{example}

\end{frame}

\section{Integrable fields}

\begin{frame}{Integrabe fields}

\begin{lemma}
Let $(L,\del)/(K,\del)$ be a differential field of characteristic $p>0$.
If $u\in K$ is such that $\del^{p^k}(u)\in C(K)^{\times}$ and $y\in L$ is such
that $\del(y)=\del(u)/u$, then $\del^{p^{k+1}}(y)\in C(L)^{\times}$.
\end{lemma}

\begin{prop}\label{loglog}
Let $(K,\del)$ be a differential field of characteristic $p>0$, and
assume that there exists $X\in K$ such that $\del(X)=1$. Then for all integers
$n$ there exist an elementary differential extension $L/K$ and an element 
$y\in L$ such that $\del^n(y)=1$.
\end{prop}
\end{frame}
\begin{frame}{Integrable fields}

\begin{example}
Let $p=2$ we find for
$$\begin{array}{cl}
n = 1 & x \\
n = 2 & x^2*\log(x) \\
n = 4 & x^4*\log(x)^2*\log(\log(x)) \\
n = 8 & x^8*\log(x)^2*\log(\log(x))^4*\log(\log(\log(x)))
\end{array}$$
with obvious notation.
\end{example}

\end{frame}
\begin{frame}{Integrable fields}

\begin{definition}
A differential field $(K,\del)$ is \emph{integrable} if it is linear and 
$\del$ takes the value $1$ on $K$.
\end{definition}

\begin{example}
The differential field $(\FF_p(X),\del)$ where $\del$ is the standard
derivation is integrable.

Indeed, every element $u$ satisfies $\del^p(u)=0$, and $\del(X)=1$.
\end{example}

\begin{theorem}
Every elementary extension of an integrable differential field is integrable.
\end{theorem}
\end{frame}

\begin{frame}{Integrable fields}

\begin{lemma}\label{logpol}
Let $(K,\del)$ be a differential field. If $u$ and $v$ in $K$ and $n\geq 1$ 
are such that $\del^n(u)=0$ and $\del^{n-1}(v)=1$, then $u$ belongs to the 
vector space generated by $(\del^k(v))_{k=0}^{n-1}$ over $C(K)$.
\end{lemma}

\begin{theorem}\label{integ}
Let $(K,\del)$ be an integrable differential field.
Every element $u\in K$ admits an antiderivative in an extension of logarithmic
type.
\end{theorem}
\end{frame}

\begin{frame}{Integrable fields}

\begin{example}
We look for the characterictic $p$ analogue of the antiderivative of $exp(exp(x))$.
From $\del^p(F)-\del(F)=E^p*F$, we conclude that 
$\frac{\del^{p-1}(F)-F}{E^p}$ is an antiderivative of $F$.
\end{example}

\end{frame}
\section{Linear differential equations}
\begin{frame}{Linear differential equations}
\begin{theorem}
Let $(K,\del)$ be an integrable differential field and $Q\in C(K)[X]$.
The linear differential equation $Q(\del)(y)=0$ admits a solution in
some elementary extension of $K$.
\end{theorem}
\begin{theorem}\label{KtoCL}
Let $(K,\del_K)$ be a linear differential field, and $(L,\del_L)$ a
differential extension of $(K,\del_K)$. If $y\in L$ statisfies
a linear differential with coefficients in $K$ then it satisfies a
linear differential equation with coefficients in $C(K)$.
\end{theorem}
\end{frame}

\begin{frame}{Linear differential equations}
\begin{theorem}
Let $(K,\del)$ be an integrable differential field and $P\in K[X]$.
The linear differential equation $P(\del)(y)=0$ admits a solution in
some elementary extension of $K$.
\end{theorem}
\end{frame}
\end{document}
