# DIFFERENTIAL CORRESPONDENCES
AND CONTROL THEORY

J.-F. Pommaret
CERMICS, Ecole des Ponts ParisTech, France
<EMAIL_ADDRESS>
ORCID:0000-0003-0907-2601

ABSTRACT

When a differential field $K$ having $n$ commuting derivations is given
together with two finitely generated differential extensions $L$ and $M$ of
$K$, an important problem in differential algebra is to exhibit a common
differential extension $N$ in order to define the new differential extensions
$L\cap M$ and the smallest differential field $(L,M)\subset N$ containing both
$L$ and $M$. Such a result allows to generalize the use of complex numbers in
classical algebra. Having now two finitely generated differential modules $L$
and $M$ over the non-commutative ring ring $D=K[d_{1},...,d_{n}]=K[d]$ of
differential operators with coefficients in $K$, we may similarly look for a
differential module $N$ containing both $L$ and $M$ in order to define $L\cap
M$ and $L+M$. This is exactly the situation met in linear or non-linear OD or
PD control theory by selecting the inputs and the outputs among the control
variables. However, in many recent books and papers, we have shown that
controllability was a built-in property of a control system, not depending on
the choice of inputs and outputs. The main purpose of this paper is to revisit
control theory by showing the specific importance of the two previous problems
and the part plaid by $N$ in both cases for the parametrization of the control
system. The essential tool will be the study of differential correspondences,
a modern name for what was called Bäcklund problem during the last century,
namely the study of elimination theory for groups of variables among systems
of linear or nonlinear OD or PD equations. The difficulty is to revisit
differential homological algebra by using non-commutative localization.
Finally, when $M$ is a $D$-module, this paper is using for the first time the
fact that the system $R=hom_{K}(M,K)$ is a $D$-module for the Spencer operator
acting on sections, avoiding thus behaviors, trajectories and signal spaces.

KEY WORDS
Differential modules; Differential extensions; Differential elimination;
Controllability;
Kähler differentials; Bäcklund problem.

1) INTRODUCTION

The story started in 1970 at Princeton university when the author of tis paper
was a visiting student of D.C. Spencer and his colleague J. Wheeler from the
nearby physics department set up a 1000 $ challenge for proving that Einstein
equations could be parametrized by potential-like functions like Maxwell
equations. It is only in 1995 that he found the negative solution of this
chalenge, only paid back one dollar (!) by Wheeler because the relativistic
community was (and still is !) convinced about the existence of such a
parametrization. Accordingly, such a result can only be found today in books
of control theory ([22],[25],[44]). Presenting this result at the algebra
seminar of M.-P. Malliavin in the Institut Henri Poincaréof Paris the same
year, he found by chance on display in the library the translation from
Japanese of the 1970 Master thesis of M. Kashiwara and discovered the
usefulness of differential homological algebra that Spencer never told him
about during his stay in Princeton ([10]).

Meanwhile, U. Oberst from Innsbruck university succeeded applying these new
tools to control theory, in particular for studying controllability for multi-
dimensional systems with constant coefficients ([14],[15]). However, the
reader may discover on the net (www.ricam.oeaw.ac.at/oberst) how difficult it
is to communicate with people familiar with analysis but not with the formal
methods (jet theory, differential sequences, diagram chasing) when studying
systems of ordinary differential (OD) or partial differential (PD) equations.
In the meantime, the author had become aware of the new methods (tensor
products of rings and fields with derivations) used by A. Bialynicki-Birula in
order to study ”Differential Galois Theory” ([2],[18]) that are largely
superseding the approach of E. Kolchin in classical differential algebra
([9],[11]).

A possibility to escape from such a situation was to publish as fast as
possible a book presenting for the first time in a self-contained way the non-
commutative aspect of double duality for the study of systems having
coefficients in a differential field $K$ ([21], Zbl 1079.93001). Of course,
the difficulty was to use commutative algebra for the graded modules in order
to study the corresponding filtred modules, a hard task indeed. A more
specific application of theses new tools to mathematical physics (general
relativity, gauge theory) allowed the author to justify the many doubts he
already had since a long time about the origin and existence of gravitational
waves and black holes, but this is out of the scope of this paper (Comparing
[1] to [32] and [35] needs no comment).

In the second section we shall study the linear framework and in the third
section we shall study the nonlinear framework, separating in each situation
the differential geometric approach from the differential algebraic approach
and providing various motivating examples. Many of the results are given
without proofs that can be found in the many books ([17-22],[24],[29],[30])
and recent papers ([24],[27],[28],[32],[33]) that we have published. It does
not seem that that the link existing between the Spencer operator and non-
commutative localization of Ore domains is known. In any case, this result has
never been used for applications to control theory or even mathematical
physics.

2) LINEAR CORRESPONDENCES

2.1) LINEAR SYSTEMS

If $X$ is a manifold of dimension $n$ with local coordinates
$(x)=(x^{1},...,x^{n})$, we denote as usual by $T=T(X)$ the tangent bundle of
$X$, by $T^{*}=T^{*}(X)$ the cotangent bundle, by ${\wedge}^{r}T^{*}$ the
bundle of r-forms and by $S_{q}T^{*}$ the bundle of q-symmetric tensors. More
generally, let $E,F,\dots$ be vector bundles over $X$ with local coordinates
$(x^{i},y^{k}),(x^{i},z^{l})),\dots$ for $i=1,...,n$, $k=1,...,m$, $l=1,...,p$
simply denoted by $(x,y)$, $(x,z)$, projection $\pi:E\rightarrow
X:(x,y)\rightarrow(x)$ and changes of coordinates
$\bar{x}=\varphi(x),\bar{y}=A(x)y$. We shall denote by $E^{*}$ the vector
bundle obtained by inverting the matrix $A$ of the changes of coordinates ,
exactly like $T^{*}$ is obtained from $T$. We denote by $\xi:X\rightarrow
E:(x)\rightarrow(x,y=\xi(x))$ a (local) section of $E$, Under a change of
coordinates, a section transforms like $\bar{\xi}(\varphi(x))=A(x)\xi(x)$ and
the changes of the derivatives can also be obtained with more work. We shall
denote by $J_{q}(E)$ the q-jet bundle of $E$ with local coordinates
$(x^{i},y^{k},y^{k}_{i},y^{k}_{ij},...)=(x,y_{q})$ called jet coordinates and
sections
${\xi}_{q}:(x)\rightarrow(x,{\xi}^{k}(x),{\xi}^{k}_{i}(x),{\xi}^{k}_{ij}(x),...)=(x,{\xi}_{q}(x))$
transforming like the sections
$j_{q}(\xi):(x)\rightarrow(x,{\xi}^{k}(x),{\partial}_{i}{\xi}^{k}(x),{\partial}_{ij}{\xi}^{k}(x),...)=(x,j_{q}(\xi)(x))$
where both ${\xi}_{q}$ and $j_{q}(\xi)$ are over the section $\xi$ of $E$. For
any $q\geq 0$, $J_{q}(E)$ is a vector bundle over $X$ with projection
${\pi}_{q}$ while $J_{q+r}(E)$ is a vector bundle over $J_{q}(E)$ with
projection ${\pi}^{q+r}_{q},\forall r\geq 0$.

DEFINITION 1.A.1: A linear system of order $q$ on $E$ is a vector sub-bundle
$R_{q}\subset J_{q}(E)$ and a solution of $R_{q}$ is a section $\xi$ of $E$
such that $j_{q}(\xi)$ is a section of $R_{q}$.

Let $\mu=({\mu}_{1},...,{\mu}_{n})$ be a multi-index with length
${\mid}\mu{\mid}={\mu}_{1}+...+{\mu}_{n}$, class $i$ if
${\mu}_{1}=...={\mu}_{i-1}=0,{\mu}_{i}\neq 0$ and
$\mu+1_{i}=({\mu}_{1},...,{\mu}_{i-1},{\mu}_{i}+1,{\mu}_{i+1},...,{\mu}_{n})$.
We set $y_{q}=\\{y^{k}_{\mu}{\mid}1\leq k\leq m,0\leq{\mid}\mu{\mid}\leq q\\}$
with $y^{k}_{\mu}=y^{k}$ when ${\mid}\mu{\mid}=0$. There is a natural way to
distinguish the section ${\xi}_{q}$ from the section $j_{q}(\xi)$ by
introducing the Spencer operator $d:J_{q+1}(E)\rightarrow T^{*}\otimes
J_{q}(E)$ with components
$(d{\xi}_{q+1})^{k}_{\mu,i}(x)={\partial}_{i}{\xi}^{k}_{\mu}(x)-{\xi}^{k}_{\mu+1_{i}}(x)$.
The kernel of $d$ consists of sections such that
${\xi}_{q+1}=j_{1}({\xi}_{q})=...=j_{q+1}(f)$. Finally, if $R_{q}\subset
J_{q}(E)$ is a system of order $q$ on $E$ locally defined by linear equations
${\Phi}^{\tau}(x,y_{q})\equiv a^{\tau\mu}_{k}(x)y^{k}_{\mu}=0$, the
$r$-prolongation $R_{q+r}={\rho}_{r}(R_{q})=J_{r}(R_{q})\cap J_{q+r}(E)\subset
J_{r}(J_{q}(E))$ is locally defined when $r=1$ by the linear equations
${\Phi}^{\tau}(x,y_{q})=0,d_{i}{\Phi}^{\tau}(x,y_{q+1})\equiv
a^{\tau\mu}_{k}(x)y^{k}_{\mu+1_{i}}+{\partial}_{i}a^{\tau\mu}_{k}(x)y^{k}_{\mu}=0$
and has symbol $g_{q+r}=R_{q+r}\cap S_{q+r}T^{*}\otimes E\subset J_{q+r}(E)$
if one looks at the top order terms. If ${\xi}_{q+1}\in R_{q+1}$ is over
${\xi}_{q}\in R_{q}$, differentiating the identity
$a^{\tau\mu}_{k}(x){\xi}^{k}_{\mu}(x)\equiv 0$ with respect to $x^{i}$ and
substracting the identity
$a^{\tau\mu}_{k}(x){\xi}^{k}_{\mu+1_{i}}(x)+{\partial}_{i}a^{\tau\mu}_{k}(x){\xi}^{k}_{\mu}(x)\equiv
0$, we obtain the identity
$a^{\tau\mu}_{k}(x)({\partial}_{i}{\xi}^{k}_{\mu}(x)-{\xi}^{k}_{\mu+1_{i}}(x))\equiv
0$ and thus the restriction $d:R_{q+1}\rightarrow T^{*}\otimes R_{q}$. More
generally, we have the restriction:

$d:{\wedge}^{s}T^{*}\otimes R_{q+1}\rightarrow{\wedge}^{s+1}T^{*}\otimes
R_{q}:({\xi}^{k}_{\mu,I}(x)dx^{I})\rightarrow(({\partial}_{i}f^{k}_{\mu,I}(x)-{\xi}^{k}_{\mu+1_{i},I}(x))dx^{i}\wedge
dx^{I})$ $None$

using standard multi-index notation for exterior forms, namely
$I=\\{i_{1}<i_{2}<...<i_{r}\\}$, $dx^{I}=dx^{i_{1}}\wedge...\wedge
dx^{i_{r}}\in{\wedge}^{r}T^{*}$ for a finite basis, and one can easily check
that $d\circ d=0$. The restriction of $-d$ to the symbol is called the Spencer
map $\delta:{\wedge}^{s}T^{*}\otimes
g_{q+1}\rightarrow{\wedge}^{s+1}T^{*}\otimes g_{q}$ and $\delta\circ\delta=0$
similarly, leading to the purely algebraic $\delta$-cohomology
$H^{s}_{q+r}(g_{q})$ ([17]-[20],[22],[30],[43]).

DEFINITION 1.A.2: A system $R_{q}$ is said to be formally integrable when all
the equations of order $q+r$ are obtained by $r$ prolongations only, $\forall
r\geq 0$ or, equivalently, when the projections
${\pi}^{q+r+s}_{q+r}:R_{q+r+s}\rightarrow R^{(s)}_{q+r}\subseteq R_{q+r}$ are
epimorphisms $\forall r,s\geq 0$.

Finding an intrinsic test has been achieved by D.C. Spencer in 1965 ([43])
along coordinate dependent lines sketched by M. Janet in 1920 ([6]) and
providing a Pommaret basis by using the Janet tabular in a particular
coordinate system called $\delta$-regular like in ([17]-[20]).

THEOREM 1.A 3: $R_{q}$ is formally integrable (involutive) if
${\pi}^{q+1}_{q}:R_{q+1}\rightarrow R_{q}$ is an epimorphism and $g_{q}$ is
$2$-acyclic with $H^{2}_{q+r}(g_{q})=0,\forall r\geq 0$ (involutive with
$H^{s}_{q+r}(g_{q})=0,\forall r\geq 0,\forall s=1,...,n$). When $R_{q}$ is
involutive, there exist $n$ integers
${\alpha}^{1}_{q}\geq...\geq{\alpha}^{n}_{q}=\alpha\geq 0$ called characters
and we have
$dim(g_{q+r})={\Sigma}^{n}_{i=1}\frac{(r+i-1)!}{r!(i-1)!}{\alpha}^{i}_{q}$, in
particular
$dim(g_{q})={\alpha}^{1}_{q}+...+{\alpha}^{n}_{q},\,\,\,dim(g_{q+1})={\alpha}^{1}_{q}+...+n{\alpha}^{n}_{q}$.

REMARK 1.A.4: As long as the Prolongation/Projection (PP) procedure has not
been achieved in order to get an involutive system $R^{(s)}_{q+r}$ for $r,s$
large enough, nothing can be said about the CC (Fine examples can be found in
[30] and the recent [32]).

When $R_{q}$ is involutive, the linear differential operator
${\cal{D}}:E\stackrel{{\scriptstyle
j_{q}}}{{\rightarrow}}J_{q}(E)\stackrel{{\scriptstyle\Phi}}{{\rightarrow}}J_{q}(E)/R_{q}=F_{0}$
of order $q$ is said to be involutive. Introducing the set of solutions
$\Theta\subseteq E$ and the Janet bundles:

$F_{r}={\wedge}^{r}T^{*}\otimes J_{q}(E)/({\wedge}^{r}T^{*}\otimes
R_{q}+\delta(S_{q+1}T^{*}\otimes E))$ $None$

we obtain the canonical linear Janet sequence (Introduced in [17], p 185 + p
391):

$0\longrightarrow\Theta\longrightarrow
E\stackrel{{\scriptstyle\cal{D}}}{{\longrightarrow}}F_{0}\stackrel{{\scriptstyle{\cal{D}}_{1}}}{{\longrightarrow}}F_{1}\stackrel{{\scriptstyle{\cal{D}}_{2}}}{{\longrightarrow}}...\stackrel{{\scriptstyle{\cal{D}}_{n}}}{{\longrightarrow}}F_{n}\longrightarrow
0$ $None$

where each other operator, induced by the Spencer operator, is first order
involutive and generates the compatibility conditions (CC) of the preceding
one. Similarly, introducing the Spencer bundles:

$C_{r}={\wedge}^{r}T^{*}\otimes R_{q}/\delta({\wedge}^{r-1}T^{*}\otimes
g_{q+1})$ $None$

we obtain the canonical linear $Spencer$ sequence also induced by the Spencer
operator:

$0\longrightarrow\Theta\stackrel{{\scriptstyle
j_{q}}}{{\longrightarrow}}C_{0}\stackrel{{\scriptstyle
D_{1}}}{{\longrightarrow}}C_{1}\stackrel{{\scriptstyle
D_{2}}}{{\longrightarrow}}...\stackrel{{\scriptstyle
D_{n}}}{{\longrightarrow}}C_{n}\longrightarrow 0$ $None$

In the case of analytic systems, the following theorem providing the Cartan-
Kähler CK) data is well known though its link with involution is rarely quoted
because it is usually presented within the framework of exterior calculus
([17],[26]):

THEOREM 1.A.5 (Cartan-Kähler): If $R_{q}\subset J_{q}(E)$ is a linear
involutive and analytic system of order $q$ on $E$, there exists one analytic
solution $y^{k}=f^{k}(x)$ and only one such that:
1) $(x_{0},{\partial}_{\mu}f^{k}(x_{0}))$ with $0\leq\mid\mu\mid\leq q-1$ is a
point of $R_{q-1}={\pi}^{q}_{q-1}(R_{q})\subset J_{q-1}(E)$.
2) For $i=1,...,n$ the ${\alpha}^{i}_{q}$ parametric derivatives
${\partial}_{\mu}f^{k}(x)$ of class $i$ are equal for
$x^{i+1}=x^{i+1}_{0},...,x^{n}=x^{n}_{0}$ to ${\alpha}^{i}_{q}$ given analytic
functions of $x^{1},...,x^{i}$.

The monomorphism $0\rightarrow J_{q+1}(E)\rightarrow J_{1}(J_{q}(E))$ allows
to identify $R_{q+1}$ with its image ${\bar{R}}_{1}$ in $J_{1}(R_{q})$ and we
just need to set $R_{q}=\bar{E}$ in order to obtain the first order system
(Spencer form) ${\bar{R}}_{1}\subset J_{1}(\bar{E})$ which is also involutive
and analytic while ${\pi}^{1}_{0}:{\bar{R}}_{1}\rightarrow\bar{E}$ is an
epimorphism. Studying the respective symbols, we may identify $g_{q+r}$ and
${\bar{g}}_{r}$ while ${\bar{g}}_{1}$ is involutive. Looking at the Janet
board of multiplicative variables we have
${\bar{\alpha}}^{i}_{1}+{\bar{\beta}}^{i}_{1}=\bar{m}=dim(\bar{E})$ and:

${\bar{\alpha}}^{i}_{1}={\alpha}^{i}_{q}+...+{\alpha}^{n}_{q}={\alpha}^{i}_{q+1}\Rightarrow{\alpha}^{i}_{q}={\bar{\alpha}}^{i}_{1}-{\bar{\alpha}}^{i+1}_{1}={\bar{\beta}}^{i+1}_{1}-{\bar{\beta}}^{i}_{1}$

We obtain therefore:

COROLLARY 1.A.6: If $R_{1}\subset J_{1}(E)$ is a first order linear involutive
and analytic system such that ${\pi}^{1}_{0}:R_{1}\rightarrow E$ is an
epimorphism, then there exists one analytic solution $y^{k}=f^{k}(x)$ and only
one, such that:
1) $f^{1}(x),...,f^{{\beta}^{1}_{1}}(x)$ are equal to ${\beta}^{1}_{1}$ given
constants when $x=x_{0}$.
2) $f^{{\beta}^{i}_{1}+1}(x),...,f^{{\beta}^{i+1}_{1}}(x)$ are equal to
${\beta}^{i+1}_{1}-{\beta}^{i}_{1}$ given analytic functions of
$x^{1},...,x^{i}$ when $x^{i+1}=x^{i+1}_{0},...,x^{n}=x^{n}_{0}$.
3) $f^{{\beta}^{n}_{1}+1}(x),...,f^{m}(x)$ are $m-{\beta}^{n}_{1}$ given
analytic functions of $x^{1},...,x^{n}$.

2.2) DIFFERENTIAL MODULES

If $A$ is an associative ring with unit $1\in A$, a subset $S\subset A$ is
called a multiplicative subset if $1\in S,0\notin S,st\in S,\forall s,t\in
S$.. In the commutative case, these conditions are sufficient to localize $A$
at $S$ by constructing the new ring of fractions $S^{-1}A$ over $A$. For
simplicity, we shall suppose that $A$ is an integral domain (no divisor of
zero) and we shall choose $S=A-\\{0\\}$ in order to introduce the field of
fractions $Q(A)=S^{-1}A=AS^{-1}$. The idea is to exhibit new quantities
written $\frac{a}{s}$ with the standard rules:

$b(\frac{a}{s})=\frac{ab}{s},\,\,\,\frac{a}{s}+\frac{b}{t}=\frac{at+bs}{st},\,\,\,\frac{a}{s}\frac{b}{t}=\frac{ab}{st}\hskip
56.9055pt\forall a,b\in A,\forall s,t\in S$

The same definition can be used for any module $M$ over $A$ in order to
introduce the module of fractions $S^{-1}M$ over $S^{-1}A$ with the rules:

$\frac{a}{s}\frac{x}{t}=\frac{ax}{st},\,\,\,\frac{x}{s}+\frac{y}{t}=\frac{tx+sy}{st},\hskip
56.9055pt\forall x,y\in M$

DEFINITION 2.2.1: $t(M)=t_{S}(M)=\\{x\in M\mid\exists s\in S,sx=0$ is called
the torsion submodule of $M$ over $S$ and we have the exact sequence
$0\rightarrow t(M)\rightarrow
M\stackrel{{\scriptstyle\theta}}{{\rightarrow}}S^{-1}M$ of modules over $A$
where the morphism $\theta$ on the right is
$x\rightarrow\frac{x}{1}=\frac{sx}{s}$ and we have
$S^{-1}M=S^{-1}A{\otimes}_{A}M$.

In the non-commutative case considered through all this paper, we shall meet
four problems:
$\bullet$ How to compare $s^{-1}a$ with $as^{-1}$ ?.
$\bullet$ How to decide when we shall say that $s^{-1}a=t^{-1}b$ ?.
$\bullet$ How to multiply $s^{-1}a$ by $t^{-1}b$ ?.
$\bullet$ How to find a common denominator for $s^{-1}a+t^{-1}b$ ?.

LEMMA 2.2.2: If there exists a left localization of a noetherian $A$ with
respect to $S$, then we must have $Sa\cap As\neq\emptyset$. It follows that
$As\cap At\cap S\neq\emptyset$ and two fractions can be multiplied or brought
to the same denominator. Finally, $t(M)$ is a submodule of $M$.

Proof: Roughly, any right fraction $as^{-1}$ can be written as a left fraction
$t^{-1}b$, that is we must have $ta=bs$. Now, if we have two fractions
$s^{-1}a$ and $t^{-1}b$, we can find $u,v\in A$ such that $us=vt\in S$. Hence,
we obtain $s^{-1}a=(us)^{-1}(ua)$ and $t^{-1}b=(vt)^{-1}vb=(us)^{-1}vb$. As
for the multiplication of fractions, we have
$(s^{-1}a)(t^{-1}b)=s^{-1}(at^{-1})b=s^{-1}(u^{-1}c)b=(us)^{-1}(cb)$.
Finally, given $x,y\in t(M)$, we can find $s,t\in S$ such that $sx=0,ty=0$. We
may thus find $u,v\in A$ such that $us=vt\in S$ and we get
$us(x+y)=usx+vty=0\Rightarrow x+y\in t(M)$. Also, we can use $ta=bs$ in order
to obtain $t(ax)=(ta)x=(bs)x=b(sx)=0\Rightarrow ax\in t(M)$.
$\Box$

Let $K$ be a differential field with $n$ commuting derivations
$({\partial}_{1},...,{\partial}_{n})$ and consider the ring
$D=K[d_{1},...,d_{n}]=K[d]$ of differential operators with coefficients in $K$
with $n$ commuting formal derivatives satisfying
$d_{i}a=ad_{i}+{\partial}_{i}a$ in the operator sense. If $P=a^{\mu}d_{\mu}\in
D=K[d]$, the highest value of ${\mid}\mu{\mid}$ with $a^{\mu}\neq 0$ is called
the order of the operator $P$ and the ring $D$ with multiplication
$(P,Q)\longrightarrow P\circ Q=PQ$ is filtred by the order $q$ of the
operators. We have the filtration $0\subset K=D_{0}\subset
D_{1}\subset...\subset D_{q}\subset...\subset D_{\infty}=D$. As an algebra,
$D$ is generated by $K=D_{0}$ and $T=D_{1}/D_{0}$ with $D_{1}=K\oplus T$ if we
identify an element $\xi={\xi}^{i}d_{i}\in T$ with the vector field
$\xi={\xi}^{i}(x){\partial}_{i}$ of differential geometry, but with
${\xi}^{i}\in K$ now. It follows that $D={}_{D}D_{D}$ is a bimodule over
itself, being at the same time a left $D$-module by the composition
$P\longrightarrow QP$ and a right $D$-module by the composition
$P\longrightarrow PQ$. We define the adjoint functor $ad:D\longrightarrow
D^{op}:P=a^{\mu}d_{\mu}\longrightarrow ad(P)=(-1)^{\mid\mu\mid}d_{\mu}a^{\mu}$
and we have $ad(ad(P))=P$ both with $ad(PQ)=ad(Q)ad(P),\forall P,Q\in D$. Such
a definition can be extended to any matrix of operators by using the
transposed matrix of adjoint operators (See
[20]-[22],[23],[24],[31],[33],[36],[37] for more details and applications to
control theory or mathematical physics).

PROPOSITION 2.2.3: $D$ is an Ore domain and $S=D-\\{0\\}\Rightarrow
S^{-1}D=DS^{-1}$.

Proof: Let $U\in S$ and $P\in D$ be given. In order to prove the Ore property
for $D$, we must find $V\in S$ and $Q\in D$ such that $VP=QU$. Considering the
system $Py=u,Uy=v$, it defines a differential module $M$ over $D$ with the
finite presentation $D^{2}\rightarrow D\rightarrow M\rightarrow 0$. Now, as we
have only one unknown and $D=Dy$ in this sequence, then $M$ is a torsion
module and $rk_{D}(M)=0$. From the additivity property of the differential
ranks, if there should be no compatibilty condition (CC), see the example
below), then the first morphism on the left should be a monomorphism, a result
leading to the contradiction $2-1+0=0$. Accordingly we can find the operators
$V$ and $Q$ such that $PU^{-1}=V^{-1}Q$. Conversely, if now $V$ and $Q$ are
given, using the adjoint functor and the fact that $ad(ad(P))=P,\forall P\in
D$, we may obtain $ad(V)$ and $ad(Q)$ such that $ad(P)ad(V)=ad(U)ad(Q)$ as
before and thus $V=ad(ad(V))$ and $Q=ad(ad(Q))$ such that $VP=QU$, a result
showing that $V^{-1}Q=PU^{-1}$.
$\Box$

EXAMPLE 2.2.4: With $m=1,n=2,q=1,K=\mathbb{Q}(x^{1},x^{2})$ let us consider
the two first order operators $U=d_{2}\in S,P=d_{1}+x^{2}$. Considering the
formal system $d_{1}y+x^{2}y=u,d_{2}y=v$, we obtain $y=d_{2}u-d_{1}v-x^{2}v$
and thus the involutive system with jet notations:

$\left\\{\begin{array}[]{lcl}y_{2}&=&v\\\ y_{1}+x^{2}y&=&u\\\
y&=&u_{2}-v_{1}-x^{2}v\end{array}\right.\framebox{ $\begin{array}[]{cc}1&2\\\
1&\bullet\\\ \bullet&\bullet\end{array}$ }$

Among the three CC that should exist, only two are non-trivial and provide the
new second order (care !) involutive sytem:

$\left\\{\begin{array}[]{lcl}A&\equiv&u_{22}-v_{12}-x^{2}v_{2}-2v=0\\\
B&\equiv&u_{12}+x^{2}u_{2}-u-v_{11}-2x^{2}v_{1}-(x^{2})^{2}v=0\end{array}\right.\framebox{
$\begin{array}[]{cc}1&2\\\ 1&\bullet\end{array}$ }$

with the unexpected single first order CC $C\equiv d_{2}B-d_{1}A-x^{2}A=0$. We
obain therefore the two operator identities:

$d_{22}(d_{1}+x^{2})=(d_{12}+x^{2}d_{2}+2)d_{2},\,\,\,\,(d_{12}+x^{2}d_{2}-1)(d_{1}+x^{2})=(d_{11}+2x^{2}d_{1}+(x^{2})^{2})d_{2}$

leading again to the two unexpected localizations:

$(d_{1}+x^{2})(d_{2})^{-1}=(d_{22})^{-1}(d_{12}+x^{2}d_{2}+2)=(d_{12}+x^{2}d_{2}-1)^{-1}(d_{11}+2x^{2}d_{1}+(x^{2})^{2})$

Taking the adjoint operators, we get in particular
$(d_{1}-x^{2})d_{22}=d_{2}(d_{12}-x^{2}d_{2}+1)$.
In order to achieve this example and explain why such methods, up to our
knowledge, have never been used for applications, it just remains to explain
the equality of these two fractions in this framework. Indeed, we obtain
easily the unique operator identity
$(d_{1}+x_{2})d_{22}=d_{2}(d_{12}+x^{2}d_{2}-1)$ provided by the last CC
$d_{2}B=(d_{1}+x^{2})A$ for $u$. Reducing to the same denominator can be done
if we use the operator identity:

$(d_{1}+x^{2})(d_{12}+x^{2}d_{2}+2)=d_{2}(d_{11}+2x^{2}d_{1}+(x^{2})^{2})$

produced by the same last CC $d_{2}B=(d_{1}+x^{2})A$ for $v$. We conclude this
example exhibiting the corresponding long exact sequence of differential
modules:

$0\rightarrow D\rightarrow D^{2}\rightarrow D^{2}\rightarrow D\rightarrow
M\rightarrow 0$

where we have successively from left to right:
$D=DC,D^{2}=DA+DB,D^{2}=Du+Dv,D=Dy$ with Euler-Poincaré characteristic
$1-2+2-1=rk_{D}(M)=0$ because $m=1$.

Accordingly, if $y=(y^{1},...,y^{m})$ are differential indeterminates, then
$D$ acts on $y^{k}$ by setting $d_{i}y^{k}=y^{k}_{i}\longrightarrow
d_{\mu}y^{k}=y^{k}_{\mu}$ with $d_{i}y^{k}_{\mu}=y^{k}_{\mu+1_{i}}$ and
$y^{k}_{0}=y^{k}$. We may therefore use the jet coordinates in a formal way as
in the previous section. Therefore, if a system of OD/PD equations is written
in the form ${\Phi}^{\tau}\equiv a^{\tau\mu}_{k}y^{k}_{\mu}=0$ with
coefficients $a\in K$, we may introduce the free left differential module
$Dy=Dy^{1}+...+Dy^{m}\simeq D^{m}$ and consider the differential module of
equations $I=D\Phi\subset Dy$, both with the residual left differential module
$M=Dy/D\Phi$ or $D$-module and we may set $M={}_{D}M\in mod(D)$ if we want to
specify the action of the ring of differential operators. We may introduce the
formal prolongation with respect to $d_{i}$ by setting
$d_{i}{\Phi}^{\tau}\equiv
a^{\tau\mu}_{k}y^{k}_{\mu+1_{i}}+({\partial}_{i}a^{\tau\mu}_{k})y^{k}_{\mu}$
in order to induce maps $d_{i}:M\longrightarrow
M:{\bar{y}}^{k}_{\mu}\longrightarrow{\bar{y}}^{k}_{\mu+1_{i}}$ by residue with
respect to $I$ if we use to denote the residue $Dy\longrightarrow
M:y^{k}\longrightarrow{\bar{y}}^{k}$ by a bar like in algebraic geometry.
However, for simplicity, we shall not write down the bar when the background
will indicate clearly if we are in $Dy$ or in $M$. As a byproduct, the
differential modules we shall consider will always be finitely generated
($k=1,...,m<\infty$) and finitely presented ($\tau=1,...,p<\infty$).
Equivalently, introducing the matrix of operators
${\cal{D}}=(a^{\tau\mu}_{k}d_{\mu})$ with $m$ columns and $p$ rows, we may
introduce the morphism
$D^{p}\stackrel{{\scriptstyle{\cal{D}}}}{{\longrightarrow}}D^{m}:(P_{\tau})\longrightarrow(P_{\tau}{\Phi}^{\tau})$
over $D$ by acting with $D$ on the left of these row vectors while acting with
${\cal{D}}$ on the right of these row vectors by composition of operators with
$im({\cal{D}})=I$. The presentation of $M$ is defined by the exact cokernel
sequence
$D^{p}\stackrel{{\scriptstyle{\cal{D}}}}{{\longrightarrow}}D^{m}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}M\longrightarrow 0$. We notice that the presentation
only depends on $K,D$ and $\Phi$ or $\cal{D}$, that is to say never refers to
the concept of (explicit local or formal) solutions. It follows from its
definition that $M$ can be endowed with a quotient filtration obtained from
that of $D^{m}$ which is defined by the order of the jet coordinates $y_{q}$
in $D_{q}y$. We have therefore the inductive limit $0\subseteq M_{0}\subseteq
M_{1}\subseteq...\subseteq M_{q}\subseteq...\subseteq M_{\infty}=M$ with
$d_{i}M_{q}\subseteq M_{q+1}$ and $M=DM_{q}$ for $q\gg 0$ with prolongations
$D_{r}M_{q}\subseteq M_{q+r},\forall q,r\geq 0$.

DEFINITION 2.2.5: An exact sequence of morphisms finishing at $M$ is said to
be a resolution of $M$. If the differential modules involved apart from $M$
are free, that is isomorphic to a certain power of $D$, we shall say that we
have a free resolution of $M$. In the general situation of a sequence
$M^{\prime}\stackrel{{\scriptstyle
f}}{{\longrightarrow}}M\stackrel{{\scriptstyle g}}{{\longrightarrow}}M"$ of
modules which may not be exact, we may define the coboundary, cocycle and
cohomology at $M$ by setting respectively $B=im(f)\subseteq
Z=ker(g)\Rightarrow H=Z/B$ and apply the above result to the various short
exact sequences like $0\rightarrow Z\rightarrow M\rightarrow im(g)\rightarrow
0$ or $0\rightarrow B\rightarrow Z\rightarrow H\rightarrow 0$. The deleted
complexis obtained by replacing $M$ by $0$. Applying $hom_{D}(\bullet,D)$, we
obtaine a sequence that may not be exact. The corresponding cohomology
modules, called extension modules $ext^{i}_{D}(M,D)=ext^{i}(M)$, are torsion
modules for $i\geq 1$, do not depend on the resolution of $M$ and only depend
on $K,D$ and $M$ ([3],[10],[21],[36],[38],[40]).

Having in mind that $K$ is a left $D$-module with the action $(D,K)\rightarrow
K:(P,a)\rightarrow P(a)$ defined by $(b,a)\rightarrow
ba=ab,(d_{i},a)\rightarrow d_{i}(a)={\partial}_{i}a$ and that $D$ is a
bimodule over itself for the composition law of operators, we have only two
possible constructions only depending on $K,D$ and $M$:

DEFINITION 2.2.6: We may define the right (care !) differential module
$hom_{D}(M,D)$, using the bimodule structure of ${}_{D}D_{D}$ and setting
$(fP)(m)=f(m)P$ while checking that:

$((fP)Q)(m)=((fP)(m))Q=(f(m)P)Q=f(m)PQ=(fPQ)(m)$

REMARK 2.2.7: When $M$ admits the finite presentation
$D^{p}\stackrel{{\scriptstyle{\cal{D}}}}{{\longrightarrow}}D^{m}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}M\rightarrow 0$, applying $hom_{D}(\bullet,D)$ we obtain
the following long exact sequence of right (care !) differential modules:

$0\leftarrow N_{D}\longleftarrow D^{p}_{D}\longleftarrow
D^{m}_{D}\longleftarrow hom_{D}(M,D)\leftarrow 0$

and the so-called Malgrange isomorphism just amounts to the fact that
$ext^{0}(M)=hom_{D}(M,D)$. We are immediately facing one of the most delicate
problems of this section when dealing with applications and/or effective
computations, a problem not solved in the corresponding literature which has
been almost entirely using fields of constants ([14],[15],[41],[42]) though
the solution is known since a long time ([3],[20],[38]). Indeed, apart for
purely mathematical reasons, the only differential modules to be met are left
differential modules. By chance, one has the following theorem describing the
functorial side changing procedure amounting to replace ${\cal{D}}$ by is
formal adjoint $ad({\cal{D}})$. In the differential geometric framework, such
a procedure amounts to replace an operator ${\cal{D}}:E\rightarrow F$ by its
formal adjoint $ad({\cal{D}}):{\wedge}^{n}T^{*}\otimes
F^{*}\rightarrow{\wedge}^{n}T^{*}\otimes E^{*}$ and one may set
$ad(E)={\wedge}^{n}T^{*}\otimes E^{*}$ for any vector bundle $E$ over $X$ with
$dim(E)=dim(ad(E))$ while reversing the arrows. However, as the formal adjoint
of an involutive operator may not even be formally integrable, the formal
adjoint of an exact (Janet, Spencer) sequence may not be an exact (Janet,
Spencer) sequence at all and this is the motivation for introducing the
extension modules. The simplest example when $n=1$ can be found in the study
of the double pendulum ([36]). In the time-varying case with $x=t$ and
$d=d_{t}$, the Kalman-type system $dy=A(x)y+B(x)u$ is controllable if and only
if the operator $\lambda\rightarrow(d\lambda+\lambda A,\lambda B)$ is
injective, that is to say if and only if the matrix
$(B,(d-A)B,(d-A)^{2}B,...)$ has maximum rank ( Compare to [45] where the
adjoint is missing).

THEOREM 2.2.8: There exists an isomorphism $N_{D}\rightarrow
N={}_{D}N=hom_{K}({\wedge}^{n}T^{*},N_{D})$ with inverse $M={}_{D}M\rightarrow
M_{D}={\wedge}^{n}T^{*}{\otimes}_{K}M$ .

Proof: First of all, we prove that ${\wedge}^{n}T^{*}$ has a natural right
module structure over $D$ by introducing the basic volume $n$-form
$\alpha=dx^{1}\wedge\dots\wedge dx^{n}=dx$ and defining
$\alpha.P=ad(P)(1)dx,\forall P\in D$. We have
$\alpha.\xi=-{\partial}_{i}{\xi}^{i}dx=-div(\xi)dx=-{\cal{L}}(\xi)\alpha,\forall\xi={\xi}^{i}d_{i}\in
T$ where ${\cal{L}}$ is the classical Lie derivative on forms and obtain
therefore:

$\alpha.(a\xi)=(a\,dx).\xi=-{\partial}_{i}(a{\xi}^{i})dx=-a{\partial}_{i}{\xi}^{i}dx-\xi(a)dx=-a{\partial}_{i}{\xi}^{i}dx-\alpha.(\xi(a))$
$\alpha.(\xi a)=(\alpha.\xi)a=-a{\partial}_{i}{\xi}^{i}dx\hskip
28.45274pt\Rightarrow\hskip 28.45274pt\alpha.({\xi\,a})=\alpha.(a\xi+\xi(a))$

From well known properties of the Lie derivative, we have also:

$\alpha.(\xi\eta-\eta\xi)=-({\cal{L}}(\xi){\cal{L}}(\eta)-{\cal{L}}(\eta){\cal{L}}(\xi))\alpha=-{\cal{L}}([\xi,\eta])\alpha$

Now, using the adjoint map $ad:D\rightarrow D:P\rightarrow ad(P)$, we may
introduce the adjoint functor $ad:mod(D)\rightarrow mod(D^{op}):M\rightarrow
ad(M)$ with $m.P=ad(P)m,\forall m\in M,\forall P\in D$ and we have
$m.(PQ)=ad(PQ)m=ad(Q)ad(P)m=(m.P).Q,\forall P,Q\in D$.
It remains to introduce the $K$-linear isomorphism $ad(M)\simeq
M_{D}:m\rightarrow m\otimes\alpha$ with:

$(m\otimes\alpha)a=am\otimes\alpha=m\otimes a\alpha,\hskip
28.45274pt(m\otimes\alpha)\xi=-\xi
m\otimes\alpha+m\otimes\alpha.\xi,\,\,\,\forall m\in M,\forall\xi\in T$

and check that $(m\otimes\alpha)(\xi a)=(m\otimes\alpha)(a\xi+\xi(a))$. These
definition are coherent because, when $d$ is any $d_{i}$, we have $div(d)=0$
and thus $(m\otimes\alpha)d=-dm\otimes\alpha$, a result leading to the
formula:

$(m\otimes\alpha)P=(m\otimes\alpha)(a^{\mu}d_{\mu})=(a^{\mu}m\otimes\alpha)d_{\mu}=(-1)^{\mid\mu\mid}d_{\mu}a^{\mu}m\otimes\alpha=ad(P)m\otimes\alpha$

The isomorphism $ad(D)\simeq D_{D}:P\rightarrow ad(P)$ is also right
$D$-linear because we have successively:

$P.Q\rightarrow ad(P.Q)=ad(ad(Q)P)=ad(P)ad(ad(Q))=ad(P)Q$

These unexpected results explain why the formal adjoint cannot be avoided in
non-commutative localization and is so important for applications ranging from
control theory ([21],[22],[27]) to continuum mechanics ([5],[23]) or
electromagnetism ([31]) and even general relativity ([33],[37]).
$\Box$

DEFINITION 2.2.9: We define the system $R=hom_{K}(M,K)$ and set
$R_{q}=hom_{K}(M_{q},K)$ as the system of order $q$. We have the projective
limit $R=R_{\infty}\longrightarrow...\longrightarrow
R_{q}\longrightarrow...\longrightarrow R_{1}\longrightarrow R_{0}$. It follows
that $f_{q}\in R_{q}:y^{k}_{\mu}\longrightarrow f^{k}_{\mu}\in K$ with
$a^{\tau\mu}_{k}f^{k}_{\mu}=0$ defines a section at order $q$ and we may set
$f_{\infty}=f\in R$ for a section of $R$. For an arbitrary differential field
$K$, such a definition has nothing to do with the concept of a formal power
series solution (care).

Similarly to the preceding definition, we may define the left (care !)
differential module $hom_{K}(D,K)$, using again the bimodule structure of $D$
and setting $(Qf)(P)=f(PQ),\forall P,Q\in D$, in particular with $(\xi
f)(P)f(P\xi),\forall\xi\in T,\forall P\in D$. However, we should have
$(af)(P)=f(Pa)\neq f(aP)=a(f(P)),\forall a\in$, unless $K$ is a field of
constants like in most of the literature ([14],[15],[42]).

PROPOSITION 2.2.10: When $M$ is a left $D$-module, then $R$ is also a left
$D$-module.

Proof: As $D$ is generated by $K$ and $T$ as we already said, let us define:

$(af)(m)=af(m)=f(am),\hskip 14.22636pt\forall a\in K,\forall m\in M$ $(\xi
f)(m)=\xi f(m)-f(\xi m),\hskip 14.22636pt\forall\xi=a^{i}d_{i}\in T,\forall
m\in M$

In the operator sense, it is easy to check that
$d_{i}a=ad_{i}+{\partial}_{i}a$ and that $\xi\eta-\eta\xi=[\xi,\eta]$ is the
standard bracket of vector fields. Using simply $\partial$ in place of any
${\partial}_{i}$ and $d$ in place of any $d_{i}$, we have:

$\begin{array}[]{rcl}((da)f)(m)=(d(af))(m)&=&d(af(m))-af(dm)\\\ &=&(\partial
a)f(m)+ad(f((m))-af(dm)\\\ &=&(a(df))(m)+(\partial a)f(m)\\\ &=&((ad+\partial
a)f)(m)\end{array}$

We finally get
$(d_{i}f)^{k}_{\mu}=(d_{i}f)(y^{k}_{\mu})={\partial}_{i}f^{k}_{\mu}-f^{k}_{\mu+1_{i}}$
and thus recover exactly the Spencer operator of the previous section though
this is not evident at all. We also get
$(d_{i}d_{j}f)^{k}_{\mu}={\partial}_{ij}f^{k}_{\mu}-{\partial}_{i}f^{k}_{\mu+1_{j}}-{\partial}_{j}f^{k}_{\mu+1_{i}}+f^{k}_{\mu+1_{i}+1_{j}}\Longrightarrow
d_{i}d_{j}=d_{j}d_{i},\forall i,j=1,...,n$ and thus $d_{i}R_{q+1}\subseteq
R_{q}\Longrightarrow d_{i}R\subset R$ induces a well defined operator
$R\longrightarrow T^{*}\otimes R:f\longrightarrow dx^{i}\otimes d_{i}f$. This
operator has been first introduced, up to sign, by F.S. Macaulay as early as
in $1916$ but this is still not acknowledged ([12]). For more details on the
Spencer operator and its applications, the reader may look at
([19],[24],[30],[29]).
$\Box$

PROPOSITION 2.2.11: When $M$ and $N$ are left $D$-modules, then
$M{\otimes}_{K}N$ is also a left $D$-module.

Proof: As before, we may define:

$a(m\otimes n)=am\otimes n=m\otimes an,\hskip 14.22636pt\forall a\in K,\forall
m\in M,\forall n\in N$ $\xi(m\otimes n)=\xi m\otimes n+m\otimes\xi n,\hskip
14.22636pt\forall\xi\in T,\forall m\in M,\forall n\in N$

and let the reader finish as an exercise.
$\Box$

COROLLARY 2.2.12: The two structures of left $D$-modules obtained in these two
propositions are coherent with the following adjoint isomorphism existing for
any triple $L,M,N\in mod(D)$:

$hom_{D}(M{\otimes}_{K}N,L)\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}hom_{D}(M,hom_{K}(N,L))$

Proof: Whenever $f\in hom_{D}(M{\otimes}_{K}N,L)$, we may define
${\varphi}(f)=g$ by $(g(m))(n)=f(m\otimes n)\in L$ and we have successively
for any $\xi\in T$:

$(\xi(g(m)))(n)=\xi((g(m))(n))-(g(m))(\xi n)=\xi(f(m\otimes n))-f(m\otimes\xi
n)=f(\xi(m\otimes n))-f(m\otimes\xi n)$

that is $(\xi(g(m)))(n)=f(\xi m\otimes n)=(g(\xi m))(n)$ and thus
$\xi(g(m))=g(\xi m),\forall m\in M$.
The inverse morphism can be studied similarly.
$\Box$

COROLLARY 2.2.13 : $R=hom_{K}(M,K)\simeq hom_{D}(M,hom_{K}(D,K))$.

Proof: As $K$ is a field, thus a commutative ring, we have the isomorphism of
left $D$-modules $M{\otimes}_{K}N\simeq N{\otimes}_{K}M:m\otimes n\rightarrow
n\otimes m,\forall m\in M,\forall n\in N$ and we may exchange $M$ and $N$. As
$K$ is a left differential module for the rule $(P,a)\rightarrow P(a),\forall
P\in D,\forall a\in K$, we obtain:

$\begin{array}[]{rcl}hom_{D}(M,(hom_{K}(D,K))\simeq
hom_{D}(M{\otimes}_{K}D,K)&\simeq&hom_{D}(D{\otimes}_{K}M,K)\\\
&\simeq&hom_{D}(D,hom_{K}(M,K))\\\ &\simeq&hom_{K}(M,K)\end{array}$

$\Box$

COROLLARY 22.14: The differential module $hom_{K}(D,K)$ is an injective
differential module.

Proof: When $0\rightarrow M^{\prime}\rightarrow M\rightarrow M"\rightarrow 0$
is a short exact sequence of modules and $N$ is any module, we have only the
exact sequence $0\rightarrow hom(M",N)\rightarrow hom(M,N)\rightarrow
hom(M^{\prime},N)$ obtained by composition of morphisms. In the present
situation, using the previous corollaries, we have the following commutative
and exact diagram because $K$ is a field (See [4], p 18):

$\begin{array}[]{cccl}0&&0&\\\ \downarrow&&\downarrow&\\\
hom_{D}(M,hom_{K}(D,K))&\rightarrow&hom_{D}(M^{\prime},hom_{K}(D,K))&\\\
\downarrow&&\downarrow&\\\
hom_{K}(M,K)&\rightarrow&hom_{K}(M^{\prime},K)&\rightarrow 0\\\
\downarrow&&\downarrow&\\\ 0&&0&\end{array}$

Chasing in this diagram, we deduce that the upper morphism is an epimorphism
and $hom_{K}(D,K)$ is an injective module because $hom(\bullet,hom_{K}(D,K))$
transforms a short exact sequence into a short exact sequence. The reader may
compare such an approach with the one used in ([41] or [42]) in order to
understand why these applications are not dealing with variable coefficients
as the differential structure on $R$ must be defined by the Spencer operator
but we do not know any other reference (compare to [14]).
$\Box$

DEFINITION 2.2.15: With any differential module $M$ we shall associate the
graded module $G=gr(M)$ over the polynomial ring $gr(D)\simeq K[\chi]$ by
setting $G={\oplus}^{\infty}_{q=0}G_{q}$ with $G_{q}=M_{q}/M_{q+1}$ and we get
$g_{q}=G_{q}^{*}$ where the symbol $g_{q}$ is defined by the short exact
sequences:

$0\longrightarrow M_{q-1}\longrightarrow M_{q}\longrightarrow
G_{q}\longrightarrow 0\hskip 11.38109pt\Longleftrightarrow\hskip
11.38109pt0\longrightarrow g_{q}\longrightarrow R_{q}\longrightarrow
R_{q-1}\longrightarrow 0$

We have the short exact sequences $0\longrightarrow D_{q-1}\longrightarrow
D_{q}\longrightarrow S_{q}T\longrightarrow 0$ leading to $gr_{q}(D)\simeq
S_{q}T$ and we may set as usual $T^{*}=hom_{K}(T,K)$ in a coherent way with
differential geometry.

The two following definitions, which are well known in commutative algebra,
are also valid (with more work) in the case of differential modules (See [21]
for more details or the references ([13],[22],[39],[40]) for an introduction
to homological algebra and diagram chasing).

DEFINITION 2.2.16: The set of elements $t(M)=\\{m\in M\mid\exists 0\neq P\in
D,Pm=0\\}\subseteq M$ is a differential module called the torsion submodule of
$M$. More generally, a module $M$ is called a torsion module if $t(M)=M$ and a
torsion-free module if $t(M)=0$. In the short exact sequence $0\rightarrow
t(M)\rightarrow M\rightarrow M^{\prime}\rightarrow 0$, the module $M^{\prime}$
is torsion-free. Its defining module of equations $I^{\prime}$ is obtained by
adding to $I$ a representative basis of $t(M)$ set up to zero and we have thus
$I\subseteq I^{\prime}$.

DEFINITION 2.2.17: A differential module $F$ is said to be free if $F\simeq
D^{r}$ for some integer $r>0$ and we shall define $rk_{D}(F)=r$. If $F$ is the
biggest free differential module contained in $M$, then $M/F$ is a torsion
differential module and $hom_{D}(M/F,D)=0$. In that case, we shall define the
differential rank of $M$ to be $rk_{D}(M)=rk_{D}(F)=r$. Accordingly, if $M$ is
defined by a linear involutive operator of order $q$, then
$rk_{D}(M)={\alpha}^{n}_{q}$.

PROPOSITION 2.2.18: If $0\rightarrow M^{\prime}\rightarrow M\rightarrow
M"\rightarrow 0$ is a short exact sequence of differential modules and maps or
operators, we have $rk_{D}(M)=rk_{D}(M^{\prime})+rk_{D}(M")$.

REMARK 2.2.19: We emphasize once more that the left $D$-module $hom_{K}(D,K)$
used in the literature ([12,[14]°541],[42],[45]) is coming from the right
action of $D$ on any $f\in hom_{K}(D,K)$ through the formula $(Qf)(P)=f(PQ)$
and we have thus:

$(af)(P)=f(Pa),\hskip 28.45274pt(\xi f)(P)=f(P\xi),\hskip 28.45274pt\forall
a\in K,\forall\xi\in T,\forall P\in D$

As $hom_{K}(M,K)\simeq hom_{D}(M,hom_{K}(D,K))$, they are not coherent at all
with the formulas of Proposition 2.2.1, namely:

$(af)(m)=a(f(m))=f(am),\hskip 28.45274pt(\xi f)(m)=\xi(f(m))-f(\xi m),\hskip
28.45274pt\forall a\in K,\forall\xi\in T,\forall m\in M$

unless $K$ is a field of constants, in particular because, when $M=D$, then
$Pa\neq aP$ in general ([30], p 66 for more details). Accordingly, most of the
applications of differential duality to control theory must be therefore
revisited with these new methods of differential homological algebra (Compare
to [45]). We also claim that the use of the adjoint operator must become
essential for all the applications to mathematical physics.
.

In order to conclude this section, we may say that the main difficulty met
when passing from the differential framework to the algebraic framework is the
” inversion ” of arrows. Indeed, with $dim(E)=m,dim(F)=p$, when an operator
${\cal{D}}:E\rightarrow F$ is injective, that is when we have the exact
sequence $0\rightarrow
E\stackrel{{\scriptstyle{\cal{D}}}}{{\longrightarrow}}F$, like in the case of
the operator $0\rightarrow E\stackrel{{\scriptstyle
j_{q}}}{{\longrightarrow}}J_{q}(E)$, on the contrary, using differenial
modules, we have the epimorphism
$D^{p}\stackrel{{\scriptstyle{\cal{D}}}}{{\longrightarrow}}D^{m}\rightarrow
0$. The case of a formally surjective operator, like the $div$ operator,
described by the exact sequence
$E\stackrel{{\scriptstyle{\cal{D}}}}{{\longrightarrow}}F\rightarrow 0$ is now
providing the exact sequence of differential modules $0\rightarrow
D^{p}\stackrel{{\scriptstyle{\cal{D}}}}{{\longrightarrow}}D^{m}\rightarrow
M\rightarrow 0$ because ${\cal{D}}$ has no CC.

We are now ready for using the results of the second section on the Cartan-
Kähler theorem. For such a purpose, separating the parametric jets that can be
chosen arbitrarily from the principal jets that can be obtained by using the
fact that the given OD or PD equations have coefficients in a differential
field $K$, we may write the solved equations in the symbolic form
$y_{pri}-c^{par}_{pri}y_{par}=0$ with $c\in K$ and an implicit (finite)
summation in order to obtain for the sections
$f_{pri}-c^{par}_{pri}f_{par}=0$. Using the language of Macaulay, it follows
that the so-called modular equations are $E\equiv
f_{pri}a^{pri}+f_{par}a^{par}=0$ with eventually an infinite number of terms
in the implicit summations. Substituting, we get at once $E\equiv
f_{par}(a^{par}+c^{par}_{pri}a^{pri})=0$. Ordering the $y_{par}$ as we already
did and using a basis $\\{(1,0,...),(0,1,0,...),(0,0,1,0,...),...\\}$ for the
$f_{par}$, we may select the parametric modular equations $E^{par}\equiv
a^{par}+c^{par}_{pri}a^{pri}=0$.

When $k$ is a field of constants, a polynomial $P=a^{\mu}{\chi}_{\mu}\in
k[\chi]$ of degree $q$ is multiplied by a monomial ${\chi}_{\nu}$ with
$\mid\nu\mid=r$, we get ${\chi}_{\nu}P=a^{\mu}{\chi}_{\mu+\nu}$. Hence, if
$0\leq\mid\mu\mid\leq q$, the ”shifted ” polynomial thus obtained is such that
$r\leq\mid\mu+\nu\mid\leq q+r$ and the difference between the maximum degree
and the minimum degree of the monomials involved is always equal to $q$ and
thus fixed. When $n=1$, one can exhibit a series only made with $0$ or $1$
like $f=(1,0,1,0,0,1,0,0,0,1,0,0,...)$ with ” zero zones ” of successive
increasing lengths $1,2,3,4,...$ and so on, separated by $1$ in such a way
that the contraction with the shifted polynomial is the leading term of the
given polynomial and extend this procedure to $n$ arbitrary.

Replacing ${\chi}_{i}$ by $d_{i}$ and degree by order, we may use the results
of section 2 in order to split the CK-data into $m$ formal power series of $0$
(constants), $1,...,n$ variables that we shall call series of type $i$ for
$i=0,1,...,n$. However, as the following elementary example will show, the
shifting procedure cannot be applied to the variable coefficient case, namely
when $K$ is used in place of $k$. Indeed, with $n=1,d_{x}=d,K=\mathbb{Q}(x)$
and $P=d^{2}-\frac{x}{3}$, if we contract
$d^{3}P=d^{5}-\frac{x}{3}d^{3}-d^{2}$ with the series $f=(1,0,1,0,0,1,0,...)$
already defined when $n=1$, we get $1-1=0$ though $P$ does not kill $f$
because $d^{2}f=(1,0,0,1,0,0,...)\Rightarrow Pf=(1-\frac{x}{3},...)$ and the
contraction of $P$ with $f$ is $1-\frac{x}{3}\neq 0$.

WE SHALL ESCAPE FROM THIS DIFFICULTY BY MEANS OF A TRICK BASED ON A SYSTEMATIC
USE OF THE SPENCER OPERATOR (Compre to [14]).

The idea will be to shift the series to the left (decreasing ordering), up to
sign, instead of shifting the operator to the right (increasing ordering). For
this, we notice that we want that the contraction of $P=a^{\mu}d_{\mu}$ where
$\mid\mu\mid\leq q=ord(P)$ with $f$ should be zero, that is
$a^{\mu}f_{\mu}=0\Rightarrow({\partial}_{i}a^{\mu})f_{\mu}+a^{\mu}({\partial}_{i}f_{\mu})=0,\forall
i=1,...,n$. But $d_{i}P=a^{\mu}d_{\mu+1_{i}}+({\partial}_{i}a^{\mu})d_{\mu}$
must also contract to zero wih $f$ that is
$a^{\mu}f_{\mu+1_{i}}+({\partial}_{i}a^{\mu})f_{\mu}=0$. Substracting, we
obtain therefore the condition
$a^{\mu}({\partial}_{i}f_{\mu}-f_{\mu+1_{i}})=0$, that is $P$ must also
contract to zero with the shift $d_{i}f$ or even $d_{\nu}f$ of $f$ when $f$ is
made with $0$ and $1$ only. Applying this computation to the above example, we
get $-df=(0,1,0,0,1,0,...)\Rightarrow
d^{2}f=(1,0,0,1,0,...)\Rightarrow-d^{3}f=(0,0,1,0,...)$ and the contraction
with $P$ provides the leading coefficient $1\neq 0$ of $P$ like the
contraction of $d^{3}P$ with $d^{4}f=(0,1,0,0,0,1,0,...)$, that is the same
series can be used but in a quite different framework. Also, in the finite
dimensional case existing when the symbol $g_{q}$ of $R_{q}$ is finite type,
that is when $g_{q+r}=0$ for a certain integer $r\geq 0$, applying the
$\delta$-sequence inductively to $g_{q+n+i}$ for $i=r-1,...,0$ as in ([17],
Proposition 4.7, p 123), it is known that $g_{q}$ is finite type and
involutive if and only if $g_{q}=0$, that is to say $dim(R_{q})=dim(R_{q-1})$.
In a coherent way, we have thus obtained: ÊÊ

THEOREM 2.2.20: If $M$ is a differential module over $D=K[d]$ defined by a
first order involutive system in the $m$ unknowns $y^{1},...,y^{m}$ with no
zero order equation, the differential module $R=hom_{K}(M,K)$ may be generated
over $D$ by a finite basis of sections containing $m$ generators.

In the general situation, counting the number of CK data, we have
${\alpha}^{1}_{q}+...+{\alpha}^{n}_{q}=dim(g_{q})$ and
$dim(R_{q})=dim(g_{q})+dim(R_{q-1})$. We obtain therefore the following result
which is coherent with the number of unknowns in the Spencer form
$R_{q+1}\subset J_{1}(R_{q})$.

COROLLARY 2.2.21: If $M$ is a differential module over $D=K[d]$ defined by an
involutive system $R_{q}\subset J_{q}(E)$, the differential module
$R=hom_{K}(M,K)$ may be generated over $D$ by a finite basis of sections
containing $dim(R_{q})$ generators.

EXAMPLE 2.2.22: Among the most interesting examples with
$m=1,n=3,K=\mathbb{Q}$ we present the following second order system provided
by Macaulay in $1916$ ([ M], p ):

$y_{33}=0,\hskip 56.9055pty_{23}-y_{11}=0,\hskip 56.9055pty_{22}=0$

It is easy to check that $g_{2}$ with $dim(g_{2})=3$ is not involutive, that
$g_{3}$ with $dim(g_{3})=1$ is $2$-acyclic because $y_{123}-y_{111}=0$ and
that $g_{4}=0$ is trivially involutive. Accordingly, only the system $R_{4}$
is involutive with the $8=2^{3}$ parametric jets
$(y,y_{1},y_{2},y_{3},y_{11},y_{12},y_{13},y_{111})$ and all the three
characters vanish both with all the jets of order $\geq 4$ because the system
is homogeneous.
Let us consider the following $8$ sections with all other components equal to
zero:

$\begin{array}[]{|c|c|c|c|c|c|c|c|c|c|c|}section&y&y_{1}&y_{2}&y_{3}&y_{11}&y_{12}&y_{13}&y_{23}&y_{111}&y_{123}\\\
\hline\cr f_{1}&1&0&0&0&0&0&0&0&0&0\\\ f_{2}&0&1&0&0&0&0&0&0&0&0\\\
f_{3}&0&0&1&0&0&0&0&0&0&0\\\ f_{4}&0&0&0&1&0&0&0&0&0&0\\\
f_{5}&0&0&0&0&1&0&0&1&0&0\\\ f_{6}&0&0&0&0&0&1&0&0&0&0\\\
f_{7}&0&0&0&0&0&0&1&0&0&0\\\ f_{8}&0&0&0&0&0&0&0&0&1&1\end{array}$

Taking into account the two PD equations $y_{23}-y_{11}=0$ and
$y_{123}-y_{111}=0$, we obtain successively:

$d_{1}f_{8}=-f_{5},\,\,d_{1}f_{7}=-f_{4},\,\,d_{3}f_{7}=-f_{2},............,d_{1}f_{2}=-f_{1}$

Or, equivalently, working with the corresponding modular equations:

$E_{8}\equiv a^{111}+a^{123}=0\,\,\,\Rightarrow\,\,\,-d_{1}E_{8}\equiv
E_{5}\equiv a^{11}+a^{23}=0$

and so on. It follows that all the sections can be generated by the single
section $f_{8}$ and all the modular equations can be generated by the single
modular equation $E_{8}=0$, a result absolutely not evidet at first sight but
coherent with the fact that the radical of the annihilator of $M$ is the
maximal ideal $\mathfrak{m}=(d_{1},d_{2},d_{3})$.
Finally, with $m=1,n=3$ but $K=\mathbb{Q}(x^{1},x^{2},x^{3})$ an even more
striking example has been provided by M. Janet in $1920$ ([6]) with the
following second order system:

$y_{33}-x^{2}\,y_{11}=0,\hskip 56.9055pty_{22}=0$

In this case, $dim(R)=12<\infty$ but $R$ can be generated by the single
modular equation:

$E\equiv a^{12333}+x^{2}\,a^{1333}+a^{1113}=0$

because all the jets of order $>5$ vanish (See [20] and [22] for more
details).

EXAMPLE 2.2.23: If $K=\mathbb{Q}(x)$ and $n=1,m=1$, let us consider the third
order OD equation $\Phi\equiv y_{xxx}-y_{x}=0$ for which we may exhibit the
basis of sections:

$\\{f_{1}=(1,0,0,0,0,...),f_{2}=(0,1,0,1,0,1,...),f_{3}=(0,0,1,0,1,0,...)\\}$

. With $d=d_{x}$ and $\partial={\partial}_{x}$, we obtain
$df_{1}=0,df_{2}=-f_{1}-f_{3},df_{3}=-f_{2}$ and check that all the sections
can be generated by a single one, namely $f_{3}$ which describes the power
series of the solution $y=f(x)=ch(x)-1$. We have indeed $\partial f=sh(x)$ and
${\partial}^{2}f(x)=ch(x)$.
With now $m=2$, let us consider the differential module defined by the system
$y^{1}_{xx}-y^{1}=0,y^{2}_{x}=0$. Setting $y=y^{1}-y^{2}$, we successively
get:

$\begin{array}[]{rcl}y_{x}=y^{1}_{x},y_{xx}=y^{1}_{xx}=y^{1},y_{xxx}=y^{1}_{x}&\Rightarrow&y^{1}=y_{xx},y^{1}_{x}=y_{x},y^{1}_{xx}=y_{xx},y^{1}_{xxx}=y_{x},...\\\
&\Rightarrow&y^{2}=y_{xx}-y,y^{2}_{x}=0,...\end{array}$

and a differential isomorphism with the module defined by the new system
$y_{xxx}-y_{x}=0$. We have seen that the sections of the second system are
easily seen to be generated by the single section $f_{3}=(0,0,1,0,...)$, a
result leading to the only generating section
$f^{1}=1,f^{1}_{x}=0,f^{1}_{xx}=1,...,f^{2}=1,f^{2}_{x}=0,...$ of the initial
system but these sections do not describe solutions because
${\partial}f^{1}-f^{1}_{x}=0$ and ${\partial}f^{2}-f^{2}_{x}=0$ but
${\partial}f^{1}_{x}-f^{1}_{xx}=-1\neq 0$. We do not know any reference in
computer algebra dealing with sections (See [26] for more details)

EXAMPLE 2.2.24: With $n=1,m=1,q=2,K=\mathbb{Q}(x)$, let us consider the second
order OD equation $\Phi\equiv y_{xx}-xy=0$. We successively obtain by
prolongation:

$y_{xxx}-xy_{x}-y=0,y_{xxxx}-2y_{x}-x^{2}y=0,y_{xxxxx}-x^{2}y_{x}-4xy=0,y_{xxxxxx}-6xy_{x}-(x^{3}+4)y=0$

and so on. We obtain the corresponding board describing the maps
${\rho}_{r}(\Phi)$ (Compare to [M]):Ê

$\begin{array}[]{r|c|c|c|c|c|c|c|l}order&y&y_{x}&y_{xx}&y_{xxx}&y_{xxxx}&y_{xxxxx}&y_{xxxxxx}&...\\\
\hline\cr 2&-x&0&1&0&0&0&0&...\\\ 3&-1&-x&0&1&0&0&0&...\\\
4&-x^{2}&-2&0&0&1&0&0&...\\\ 5&-4x&-x^{2}&0&0&0&1&0&...\\\
6&-(x^{3}+4)&-6x&0&0&0&0&1&...\end{array}$

Let us define the sections $f^{\prime}$ and $f"$ by the following board where
$d=d_{x}$:

$\begin{array}[]{r|c|c|c|c|c|c|c|l}section&y&y_{x}&y_{xx}&y_{xxx}&y_{xxxx}&y_{xxxxx}&y_{xxxxxx}&...\\\
\hline\cr f^{\prime}&1&0&x&1&x^{2}&4x&x^{3}+4&...\\\
f"&0&1&0&x&2&x^{2}&6x&...\\\ \hline\cr
df^{\prime}&0&-x&0&-x^{2}&-2x&-x^{3}&-6x^{2}&...\\\
df"&-1&0&-x&-1&-x^{2}&-4x&-x^{3}-4&...\\\ \end{array}$

in order to obtain $df^{\prime}=-xf",df"=-f^{\prime}$. Though this is not
evident at first sight, the two boards are orthogonal over $K$ in the sense
that each row of one board contracts to zero with each row of the other though
only the rows of the first board do contain a finite number of nonzero
elements. It is absolutely essential to notice that the sections $f^{\prime}$
and $f"$ have nothing to do with solutions because $df^{\prime}\neq 0,df"\neq
0$ on one side and also because
$d^{2}f^{\prime}-xf^{\prime}=-f"=\frac{1}{x}df^{\prime}\neq 0$ even though
$d^{2}f"-xf"=0$ on the other side. As a byproduct, $f^{\prime}$ or $f"$ can be
chosen separately as unique generating section of the inverse system over $K$
(care) and we may write for example:

$f^{\prime}\rightarrow E^{\prime}\equiv
a^{0}+xa^{xx}+a^{xxx}+x^{2}a^{xxxx}+...=0,\,\,\,f"\rightarrow E"\equiv
a^{x}+xa^{xxx}+2a^{xxxx}+...=0$

EXAMPLE 2.2.25: With $n=1,m=2,q=2,K=\mathbb{Q}(x)$, let us consider again the
second order system $y^{1}_{xx}-y^{1}=0,y^{2}_{x}=0$. Setting
$z^{1}=y^{1},z^{2}=y^{1}_{x},z^{3}=y^{2}$, we obtain the first order
involutive system:

$\left\\{\begin{array}[]{lcc}z^{1}_{x}-z^{2}&=&0\\\ z^{2}_{x}-z^{1}&=&0\\\
z^{3}_{x}&=&0\end{array}\right.\framebox{$\begin{array}[]{c}x\\\ x\\\
x\end{array}$}$

It follows that the CK data for $z=g(x)$ are
$\\{g_{1}=g^{1}(0),g_{2}=g^{2}(0),g_{3}=g^{3}(0)\\}$. Using the given
equations and their solved prolongations like
$y^{1}_{xx}-y^{1}=0,y^{2}_{xx}-y^{2}=0$ and so on, we have the finite basis
(care !):

$\begin{array}[]{l|ccc|ccc|ccc|c}&z^{1}&z^{2}&z^{3}&z^{1}_{x}&z^{2}_{x}&z^{3}_{x}&z^{1}_{xx}&z^{2}_{xx}&z^{3}_{xx}&...\\\
\hline\cr g_{1}&1&0&0&0&1&0&1&0&0&...\\\ g_{2}&0&1&0&1&0&0&0&1&0&...\\\
g_{3}&0&0&1&0&0&0&0&0&0&...\end{array}$

As $dg_{1}=-g_{2},dg_{2}=-g_{1},dg_{3}=0$, a basis with only two generators
may be $\\{g^{2},g^{3}\\}$. However:

$h=g_{1}-g_{3},dh=-g_{2},d^{2}h=g_{1}\Leftrightarrow
g_{1}=d^{2}h,g_{2}=-dh,g_{3}=d^{2}h-h$

and we obtain the unique generator $h$ (See [26] for details).

2.3) LINEAR CONTROL THEORY

The most striking aspect of the application of module/system theory to linear
control theory is that it is coming from rather unexpected chases in
commutative and exact diagrams looking like rather abstract at first sight. As
more details and examples can be found in book form ([21],[22]), we shall only
provide below a few new results that cannot be found elsewhere.

PROPOSITION 2.3.1: One has the short exact sequence of (differential) modules:

$0\longrightarrow(L+M)\longrightarrow N\longrightarrow(N/L)/(M/(L\cap
M))\longrightarrow 0$

Proof: Using elementary classical homological algebra, one obtains the
following commutative and exact diagram:

$\begin{array}[]{rcccccl}&0&&0&&0&\\\ &\downarrow&&\downarrow&&\downarrow&\\\
0\rightarrow&L\cap M&\rightarrow&L&\rightarrow&L/(L\cap M)&\rightarrow 0\\\
&\downarrow&\searrow&\downarrow&&\downarrow\\\
0\rightarrow&M&\rightarrow&N&\rightarrow&N/M&\rightarrow 0\\\
&\downarrow&&\downarrow&\searrow&\downarrow&\\\ 0\rightarrow&M/(L\cap
M)&\rightarrow&N/L&\rightarrow&(N/L)/(M/(L\cap M))&\rightarrow 0\\\
&\downarrow&&\downarrow&&\downarrow\\\ &0&&0&&0&\end{array}$

At first, the lower southeast arrow being the composition of two epimorphisms
is an epimorphism. A circular chase finally proves that any element of $N$
killed by this southeast arrow is the sum of an element of $L$ and an element
of $M$, achieving the proof. It is important to notice the symmetric part
plaid by $L$ and $M$ in $N$.
$\Box$
In the general situation, we obtain from the left upper commutative square the
useful formulas:

$rk_{D}(N/(L\cap M)=rk_{D}(L/(L\cap M))+rk_{D}(N/L)=rk_{D}(M/(L\cap
M))+rk_{D}(N/M)$

in a coherent way with the following corollary:

COROLLARY 2.3.2: If $L+M=N$, then one has $L/(L\cap M)\simeq N/M$ and
$M/(L\cap M)\simeq N/L$.

THEOREM 2.3.3: There is a bijective correspondence between the intermediate
differential modules $(L\cap M)\subset L^{\prime}\subset L$ and the
intermediate differential modules $M\subset N^{\prime}\subset N$ defined by
the rules:

$L^{\prime}\rightarrow(L^{\prime}+M)=N^{\prime}\subset
N,\,\,\,\,\,N^{\prime}\rightarrow(L\cap N^{\prime})=L^{\prime}\subset L$

Proof: We have the following commutative diagram of injections:

$\begin{array}[]{ccccl}M&\rightarrow&N^{\prime}&\rightarrow&N\\\
\uparrow&&\uparrow&&\uparrow\\\ L\cap
M&\rightarrow&L^{\prime}&\rightarrow&L\end{array}$

Let us start with $L^{\prime}$, construct $N^{\prime}=L^{\prime}+M$ and obtain
$L"=L\cap N^{\prime}$. First of all, we get $L^{\prime}\subset L$ and
$L^{\prime}\subset N^{\prime}\Rightarrow L^{\prime}\subseteq L"$. Now, using
the left commutative square, we obtain from the previous proposition
$N^{\prime}/L^{\prime}\simeq M/(L\cap M)$. Similarly, using the right
commutative square, we obtain $N^{\prime}/N"\simeq N/L$ and thus an
isomorphism $N^{\prime}/L^{\prime}\simeq N^{\prime}/L"$. However, we have the
following commutative and exact diagram:

$\begin{array}[]{rcccccl}&0&&0&&0&\\\ &\downarrow&&\downarrow&&\downarrow&\\\
0\rightarrow&L^{\prime}&=&L^{\prime}&\rightarrow&0\\\
&\downarrow&&\downarrow&&\downarrow\\\
0\rightarrow&L"&\rightarrow&N^{\prime}&\rightarrow&N^{\prime}/L"&\rightarrow
0\\\ &\downarrow&&\downarrow&&\parallel&\\\
0\rightarrow&L"/L^{\prime}&\rightarrow&N^{\prime}/L^{\prime}&\rightarrow&N^{\prime}/L"&\rightarrow
0\\\ &\downarrow&&\downarrow&&\downarrow\\\ &0&&0&&0&\end{array}$

and thus $L"/L^{\prime}=0$ that is $L^{\prime}=L"$.
Finally, starting with $N^{\prime}$, we should obtain
$L^{\prime}=N^{\prime}\cap L$, then define $N"=L^{\prime}+M\subseteq
N^{\prime}$ and conclude as before that $N^{\prime}=N"$. Replacing
specialzations by injections while chasing in the following commutative and
exact diagram:

$\begin{array}[]{rcccccl}0\rightarrow&N^{\prime}&\rightarrow&N&\rightarrow&N"&\rightarrow
0\\\ &\uparrow&&\uparrow&&\uparrow&\\\
0\rightarrow&L^{\prime}&\rightarrow&L&\rightarrow&L"&\rightarrow 0\\\
&\uparrow&&\uparrow&&\uparrow&\\\ &0&&0&&0&\end{array}$

we have thus been able to deal only with submodules of $N$. In paricular, if
$N$ is torsion-free, that is $t(N)=0$, the interest of this aproach is that
all the submodules are torsion-free.

$\Box$

EXAMPLE 2.3.4: (See [21], p 736-738] for the details and diagrams) With
$n=1,K=\mathbb{Q}$, let us consider the differential module defined by the OD
equation $d^{5}v+3d^{4}v-4d^{2}v=d^{3}u-du$. We may define the input
differential module $M_{in}$ by using $u$ and the output differential module
by using $y=d^{2}v$. The differential module $(M_{in}+M_{out})\subset M$ with
a strict inclusion, is defined by the OD equation
$d^{3}y+3d^{2}y-4y=d^{3}u-du$. Localizing at the covector $\chi$, we get the
equation $(\chi-1)((\chi)+2)^{2}y=\chi(\chi-1)(\chi+1)u$ that we can also
write $(d-1)(d+2)^{2}v=d(d-1)(d+1)u$ because $K$ is a field of constants. As
we can factor by $(d-1)$ it follows that $t(M_{in}+M_{out})$ is generated by
$z=d^{2}y+4dy+4y-d^{2}u+du$ that satisfies $dz+z=0$. We have
$ann(M/M_{in})=(d^{2}(d-1)(d+2)^{2})$ and its radical is
$rad(M/M_{in})=(d(d_{1})(d+2))=(d)\cap(d-1)\cap(d+2)$ is an intersection of
prime ideals. Similarly, we have $y=0\Rightarrow d^{2}v=0,(d^{3}-d)u=0$ and
thus $ann(M/M_{out})=(d^{2})\cap(d^{3}-d)$ leading to
$rad(ann(M/M_{out}))=(d)\cap(d-1)\cap(d+1)$ as an intersection of prime
differential ideals. We have proved in ([21]) how to use these differential
submodules of $M$ both with the new differential modules
$M^{\prime}_{in}=t(M)+M_{in}$ and $M^{\prime}_{out}=t(M)+M_{out}$ in order to
study all the problems concerning poles and zeros. In the present paper, as we
are only interested by controllability, we have just to study the differential
submodules of the torsion-free differential module $M/t(M)$.

3) NONLINEAR CORRESPONDENCES

3.1) NONLINEAR SYSTEMS

If $X$ is a manifold with local coordinates $(x^{i})$ for $i=1,...,n=dim(X)$,
let us consider the fibered manifold ${\cal{E}}$ over $X$ with
${dim}_{X}({\cal{E}})=m$, that is a manifold with local coordinates
$(x^{i},y^{k})$ for $i=1,...,n$ and $k=1,...,m$ simply denoted by $(x,y)$,
projection $\pi:{\cal{E}}\rightarrow X:(x,y)\rightarrow(x)$ and changes of
local coordinates $\bar{x}=\varphi(x),\bar{y}=\psi(x,y)$. If $\cal{E}$ and
$\cal{F}$ are two fibered manifolds over $X$ with respective local coordinates
$(x,y)$ and $(x,z)$, we denote by ${\cal{E}}{\times}_{X}{\cal{F}}$ the fibered
product of $\cal{E}$ and $\cal{F}$ over $X$ as the new fibered manifold over
$X$ with local coordinates $(x,y,z)$. We denote by
$f:X\rightarrow{\cal{E}}:(x)\rightarrow(x,y=f(x))$ a global section of
$\cal{E}$, that is a map such that $\pi\circ f=id_{X}$ but local sections over
an open set $U\subset X$ may also be considered when needed. Under a change of
coordinates, a section transforms like $\bar{f}(\varphi(x))=\psi(x,f(x))$ and,
differentiating with respect to $x^{i}$, we may introduce new coordinates
$(x^{i},y^{k},y^{k}_{i})$ transforming like:

${\bar{y}}^{l}_{r}{\partial}_{i}{\varphi}^{r}(x)=\frac{\partial{\psi}^{l}}{\partial
x^{i}}(x,y)+\frac{\partial{\psi}^{l}}{\partial y^{k}}(x,y)y^{k}_{i}$

We shall denote by $J_{q}({\cal{E}})$ the q-jet bundle of $\cal{E}$ with local
coordinates $(x^{i},y^{k},y^{k}_{i},y^{k}_{ij},...)=(x,y_{q})$ called jet
coordinates and sections
$f_{q}:(x)\rightarrow(x,f^{k}(x),f^{k}_{i}(x),f^{k}_{ij}(x),...)=(x,f_{q}(x))$
transforming like the sections
$j_{q}(f):(x)\rightarrow(x,f^{k}(x),{\partial}_{i}f^{k}(x),{\partial}_{ij}f^{k}(x),...)=(x,j_{q}(f)(x))$
where both $f_{q}$ and $j_{q}(f)$ are over the section $f$ of $\cal{E}$. It
will be useful to introduce a multi-index $\mu=({\mu}_{1},...,{\mu}_{n})$ with
length $\mid\mu\mid={\mu}_{1}+...+{\mu}_{n}$ and to set
${\mu}+1_{i}=({\mu}_{1}...,{\mu}_{i-1},{\mu}_{i}+1,{\mu}_{i+1},...,{\mu}_{n})$.
Also, a jet coordinate $y^{k}_{\mu}$ is said to be of class $i$ if
${\mu}_{1}=...={\mu}_{i-1}=0,{\mu}_{i}\neq 0$. As the background will always
be clear enough, we shall use the same notation for a vector bundle or a
fibered manifold and their sets of sections [31,36]. We finally notice that
$J_{q}({\cal{E}})$ is a fibered manifold over $X$ with projection ${\pi}_{q}$
while $J_{q+r}({\cal{E}})$ is a fibered manifold over $J_{q}({\cal{E}})$ with
projection ${\pi}^{q+r}_{q},\forall r\geq 0$ [, , ].

DEFINITION 3.1.1: A (nonlinear) system of order $q$ on $\cal{E}$ is a fibered
submanifold ${\cal{R}}_{q}\subset J_{q}({\cal{E}})$ and a global or local
solution of ${\cal{R}}_{q}$ is a section $f$ of $\cal{E}$ over $X$ or
$U\subset X$ such that $j_{q}(f)$ is a section of ${\cal{R}}_{q}$ over $X$ or
$U\subset X$.

DEFINITION 3.1.2: When the changes of coordinates have the linear form
$\bar{x}=\varphi(x),\bar{y}=A(x)y$, we say that $\cal{E}$ is a vector bundle
over $X$. Vector bundles will be denoted by capital letters $E,F,\dots$ and
will have sections denoted by $\xi,\eta,\dots$. In particular, we shall denote
as usual by $T=T(X)$ the tangent bundle of $X$, by $T^{*}=T^{*}(X)$ the
cotangent bundle, by ${\wedge}^{r}T^{*}$ the bundle of r-forms and by
$S_{q}T^{*}$ the bundle of q-symmetric covariant tensors. When the changes of
coordinates have the form $\bar{x}=\varphi(x),\bar{y}=A(x)y+B(x)$ we say that
$\cal{E}$ is an affine bundle over $X$ and we define the associated vector
bundle $E$ over $X$ by the local coordinates $(x,v)$ changing like
$\bar{x}=\varphi(x),\bar{v}=A(x)v$.

DEFINITION 3.1.3: If the tangent bundle $T({\cal{E}})$ has local coordinates
$(x,y,u,v)$ changing like
${\bar{u}}^{j}={\partial}_{i}{\varphi}^{j}(x)u^{i},{\bar{v}}^{l}=\frac{\partial{\psi}^{l}}{\partial
x^{i}}(x,y)u^{i}+\frac{\partial{\psi}^{l}}{\partial y^{k}}(x,y)v^{k}$, we may
introduce the vertical bundle $V({\cal{E}})\subset T({\cal{E}})$ as a vector
bundle over $\cal{E}$ with local coordinates $(x,y,v)$ obtained by setting
$u=0$ and changes ${\bar{v}}^{l}=\frac{\partial{\psi}^{l}}{\partial
y^{k}}(x,y)v^{k}$. Of course, when $\cal{E}$ is an affine bundle over $X$ with
associated vector bundle $E$ over $X$, we have
$V({\cal{E}})={\cal{E}}\times_{X}E$. We have the short exact sequence of
vector bundles over ${\cal{E}}$:

$0\rightarrow V({\cal{E}})\rightarrow T({\cal{E}})\stackrel{{\scriptstyle
T(\pi)}}{{\longrightarrow}}{\cal{E}}{\times}_{X}T\rightarrow 0$

Accordingly, in variational calculus, the couple $(f,\delta f)$ made by a
section $f$ of ${\cal{E}}$ and its variation $\delta f$ is nothing else but a
section of $V({\cal{E}})$ while $\delta f$ has no reason at all to be ” small
”.

For a later use, if $\cal{E}$ is a fibered manifold over $X$ and $f$ is a
section of $\cal{E}$, we denote by $f^{-1}(V({\cal{E}}))$ the reciprocal image
of $V({\cal{E}})$ by $f$ as the vector bundle over $X$ obtained when replacing
$(x,y,v)$ by $(x,f(x),v)$ in each chart, along with the following commutative
diagram:

$\begin{array}[]{ccc}E&\longrightarrow&V({\cal{E}})\\\ \delta
f\uparrow\downarrow&&\downarrow\\\ X&\begin{array}[]{c}\stackrel{{\scriptstyle
f}}{{\longrightarrow}}\\\
\stackrel{{\scriptstyle\longleftarrow}}{{\pi}}\end{array}&{\cal{E}}\end{array}$

A similar construction may also be done for any affine bundle over
${\cal{E}}$. When the background is clear enough, with a slight abuse of
language, we shall sometimes set $E=V(\cal{E})$ as a vector bundle over
$\cal{E}$ and call ” vertical machinery ” such a useful systematic notation.
Looking at the transition rules of $J_{q}(\cal{E})$, we deduce easily the
following results:

PROPOSITION 3.1.4: $J_{q}(\cal{E})$ is an affine bundle over
$J_{q-1}(\cal{E})$ modeled on $S_{q}T^{*}{\otimes}_{\cal{E}}E$ but we shall
not specify the tensor product in general.

PROPOSITION 3.1.5: There is a canonical isomorphism $V(J_{q}({\cal{E}}))\simeq
J_{q}(V({\cal{E}}))=J_{q}(E)$ of vector bundles over $J_{q}(\cal{E})$ given by
setting $v^{k}_{\mu}=v^{k}_{,\mu}$ at any order and a short exact sequence:

$0\rightarrow S_{q}T^{*}\otimes E\rightarrow
J_{q}(E)\stackrel{{\scriptstyle{\pi}^{q}_{q-1}}}{{\longrightarrow}}J_{q-1}(E)\rightarrow
0$

of vector bundles over $J_{q}(\cal{E})$ allowing to establish a link with the
formal theory of linear systems.

PROPOSITION 3.1.6: There is an exact sequence:

$0\rightarrow{\cal{E}}\stackrel{{\scriptstyle
j_{q+1}}}{{\longrightarrow}}J_{q+1}({\cal{E}})\stackrel{{\scriptstyle
d}}{{\longrightarrow}}T^{*}\otimes J_{q}(E)$

where $df_{q+1}=j_{1}(f_{q})-f_{q+1}$ is over $f_{q}$ with components
$(df_{q+1})^{k}_{\mu,i}={\partial}_{i}f^{k}_{\mu}-f^{k}_{\mu+1_{i}}$ is called
the (nonlinear) Spencer operator.

DEFINITION 3.1.7: If ${\cal{R}}_{q}\subset J_{q}({\cal{E}})$ is a system of
order $q$ on ${\cal{E}}$, then
${\cal{R}}_{q+1}={\rho}_{1}({\cal{R}}_{q})=J_{1}({\cal{R}}_{q})\cap
J_{q+1}({\cal{E}})\subset J_{1}(J_{q}({\cal{E}}))$ is called the first
prolongation of ${\cal{R}}_{q}$ and we may define the subsets
${\cal{R}}_{q+r}$. In actual practice, if the system is defined by PDE
${\Phi}^{\tau}(x,y_{q})=0$ the first prolongation is defined by adding the PDE
$d_{i}{\Phi}^{\tau}\equiv{\partial}_{i}{\Phi}^{\tau}+y^{k}_{\mu+1_{i}}{\partial}{\Phi}^{\tau}/\partial
y^{k}_{\mu}=0$. accordingly,
$f_{q}\in{\cal{R}}_{q}\Leftrightarrow{\Phi}^{\tau}(x,f_{q}(x))=0$ and
$f_{q+1}\in{\cal{R}}_{q+1}\Leftrightarrow{\partial}_{i}{\Phi}^{\tau}+f^{k}_{\mu+1_{i}}(x)\partial{\Phi}^{\tau}/\partial
y^{k}_{\mu}=0$ as identities on $X$ or at least over an open subset $U\subset
X$. Differentiating the first relation with respect to $x^{i}$ and
substracting the second, we finally obtain:

$({\partial}_{i}f^{k}_{\mu}(x)-f^{k}_{\mu+1_{i}}(x))\partial{\Phi}^{\tau}/\partial
y^{k}_{\mu}=0\Rightarrow df_{q+1}\in T^{*}\otimes R_{q}$

and the Spencer operator restricts to $d:{\cal{R}}_{q+1}\rightarrow
T^{*}\otimes R_{q}$. We set
${\cal{R}}^{(1)}_{q+r}={\pi}^{q+r+1}_{q+r}({\cal{R}}_{q+r+1})$.

DEFINITION 3.1.8: The symbol of ${\cal{R}}_{q}$ is the family $g_{q}=R_{q}\cap
S_{q}T^{*}\otimes E$ of vector spaces over ${\cal{R}}_{q}$. The symbol
$g_{q+r}$ of ${\cal{R}}_{q+r}$ only depends on $g_{q}$ by a direct
prolongation procedure. We may define the vector bundle $F_{0}$ over
${\cal{R}}_{q}$ by the short exact sequence $0\rightarrow R_{q}\rightarrow
J_{q}(E)\rightarrow F_{0}\rightarrow 0$ and we have the exact induced sequence
$0\rightarrow g_{q}\rightarrow S_{q}T^{*}\otimes E\rightarrow F_{0}$ .

Setting $a^{\tau\mu}_{k}(x,y_{q})=\partial{\Phi}^{\tau}/\partial
y^{k}_{\mu}(x,y_{q})$ whenever $\mid\mu\mid=q$ and
$(x,y_{q})\in{\cal{R}}_{q}$, we obtain:

$g_{q}=\\{v^{k}_{\mu}\in S_{q}T^{*}\otimes E\mid
a^{\tau\mu}_{k}(x,y_{q})v^{k}_{\mu}=0\\},\mid\mu\mid=q,(x,y_{q})\in{\cal{R}}_{q}$
$\Rightarrow g_{q+r}={\rho}_{r}(g_{q})=\\{v^{k}_{\mu+\nu}\in
S_{q+r}T^{*}\otimes E\mid
a^{\tau\mu}_{k}(x,y_{q})v^{k}_{\mu+\nu}=0\\},\mid\mu\mid=q,\mid\nu\mid=r,(x,y_{q})\in{\cal{R}}_{q}$

In general, neither $g_{q}$ nor $g_{q+r}$ are vector bundles over
${\cal{R}}_{q}$.

On ${\wedge}^{s}T^{*}$ we may introduce the usual bases
$\\{dx^{I}=dx^{i_{1}}\wedge...\wedge dx^{i_{s}}\\}$ where we have set
$I=(i_{1}<...<i_{s})$. In a purely algebraic setting, one has:

PROPOSITION 3.1.9: There exists a map $\delta:{\wedge}^{s}T^{*}\otimes
S_{q+1}T^{*}\otimes E\rightarrow{\wedge}^{s+1}T^{*}\otimes S_{q}T^{*}\otimes
E$ which restricts to $\delta:{\wedge}^{s}T^{*}\otimes
g_{q+1}\rightarrow{\wedge}^{s+1}T^{*}\otimes g_{q}$ and
${\delta}^{2}=\delta\circ\delta=0$.

Proof: Let us introduce the family of s-forms
$\omega=\\{{\omega}^{k}_{\mu}=v^{k}_{\mu,I}dx^{I}\\}$ and set
$(\delta\omega)^{k}_{\mu}=dx^{i}\wedge{\omega}^{k}_{\mu+1_{i}}$. We obtain at
once $({\delta}^{2}\omega)^{k}_{\mu}=dx^{i}\wedge
dx^{j}\wedge{\omega}^{k}_{\mu+1_{i}+1_{j}}=0$ and
$a^{\tau\mu}_{k}(\delta\omega)^{k}_{\mu}=dx^{i}\wedge(a^{\tau\mu}_{k}{\omega}^{k}_{\mu+1_{i}})=0$.
$\Box$

The kernel of each $\delta$ in the first case is equal to the image of the
preceding $\delta$ but this may no longer be true in the restricted case and
we set:

DEFINITION 3.1.10: Let $B^{s}_{q+r}(g_{q})\subseteq Z^{s}_{q+r}(g_{q})$ and
$H^{s}_{q+r}(g_{q})=Z^{s}_{q+r}(g_{q})/B^{s}_{q+r}(g_{q})$ with
$H^{1}(g_{q})=H^{1}_{q}(g_{q})$ be the coboundary space $im(\delta)$, cocycle
space $ker(\delta)$ and cohomology space at ${\wedge}^{s}T^{*}\otimes g_{q+r}$
of the restricted $\delta$-sequence which only depend on $g_{q}$ and may not
be vector bundles. The symbol $g_{q}$ is said to be s-acyclic if
$H^{1}_{q+r}=...=H^{s}_{q+r}=0,\forall r\geq 0$, involutive if it is n-acyclic
and finite type if $g_{q+r}=0$ becomes trivially involutive for r large
enough. In particular, if $g_{q}$ is involutive and finite type, then
$g_{q}=0$. Finally, $S_{q}T^{*}\otimes E$ is involutive for any $q\geq 0$ if
we set $S_{0}T^{*}\otimes E=E$.

We have (See [17] for the diagram allowing to prove this delicate result first
found by Spencer):

PROPOSITION 3.1.11: If $g_{q}$ is $2$-acyclic and $g_{q+1}$ is a vector bundle
over ${\cal{R}}_{q}$, then $g_{q+r}$ is a vector bundle over
${\cal{R}}_{q},\forall r\geq 1$.

LEMMA 3.1.12: If $g_{q}$ is involutive and $g_{q+1}$ is a vector bundle over
${\cal{R}}_{q}$, then $g_{q}$ is also a vector bundle over ${\cal{R}}_{q}$. In
this case, changing linearly the local coordinates if necessary, we may look
at the maximum number $\beta$ of equations that can be solved with respect to
$v^{k}_{n...n}$ and the intrinsic number $\alpha=m-\beta$ indicates the number
of $y$ that can be given arbitrarily.

We notice that ${\cal{R}}_{q+r+1}={\rho}_{r}({\cal{R}}_{q+1})$ and
${\cal{R}}_{q+r}={\rho}_{r}({\cal{R}}_{q})$ in the following commutative
diagram:

$\begin{array}[]{ccc}{\cal{R}}_{q+r+1}&\stackrel{{\scriptstyle{\pi}^{q+r+1}_{q+1}}}{{\longrightarrow}}&{\cal{R}}_{q+1}\\\
\hskip 28.45274pt\downarrow{\pi}^{q+r+1}_{q+r}&&\hskip
19.91692pt\downarrow{\pi}^{q+1}_{q}\\\
{\cal{R}}^{(1)}_{q+r}&\stackrel{{\scriptstyle{\pi}^{q+r}_{q}}}{{\longrightarrow}}&{\cal{R}}^{(1)}_{q}\\\
\cap&&\cap\\\
{\cal{R}}_{q+r}&\stackrel{{\scriptstyle{\pi}^{q+r}_{q}}}{{\longrightarrow}}&{\cal{R}}_{q}\end{array}$

but we only have in general
${\cal{R}}^{(1)}_{q+r}\subseteq{\rho}_{r}({\cal{R}}^{(1)}_{q})$. We finally
obtain the following crucial Theorem and its Corollary (Compare to [17], p
70-75):

THEOREM 3.1.13: Let ${\cal{R}}_{q}\subset J_{q}({\cal{E}})$ be a system of
order $q$ on ${\cal{E}}$ such that ${\cal{R}}_{q+1}$ is a fibered submanifold
of $J_{q+1}({\cal{E}})$. If $g_{q}$ is $2$-acyclic and $g_{q+1}$ is a vector
bundle over ${\cal{R}}_{q}$, then we have
${\cal{R}}^{(1)}_{q+r}={\rho}_{r}({\cal{R}}^{(1)}_{q})$ for all $r\geq 0$.

DEFINITION 3.1.14: A system ${\cal{R}}_{q}\subset J_{q}({\cal{E}})$ is said to
be formally integrable at the order $q+r$ if
${\pi}^{q+r+s}_{q+r}:{\cal{R}}_{q+r+s}\rightarrow{\cal{R}}_{q+r}$ is an
epimorphism of fibered manifolds for all $s\geq 1$, formally integrable if
${\pi}^{q+r+1}_{q+r}$ is an epimorphism of fibered manifolds $\forall r\geq 0$
and involutive if it is formally integrable with an involutive symbol $g_{q}$.
We have the following useful test ([17],[43]):

COROLLARY 3.1.15: Let ${\cal{R}}_{q}\subset J_{q}({\cal{E}})$ be a system of
order $q$ on ${\cal{E}}$ such that ${\cal{R}}_{q+1}$ is a fibered submanifold
of $J_{q+1}({\cal{E}})$. If $g_{q}$ is $2$-acyclic (involutive) and if the map
${\pi}^{q+1}_{q}:{\cal{R}}_{q+1}\rightarrow{\cal{R}}_{q}$ is an epimorphism of
fibered manifolds, then ${\cal{R}}_{q}$ is formally integrable (involutive).

This is all what is needed in order to study nonlinear systems of ordinary
differential (OD) or partial differential (PD) equations, using calligraphic
letters like $\cal{E}$ for the nonlinear framework and capital letters like
$E=V({\cal{E}})$ for the linear or vertical linearized framework.

3.2) DIFFERENTIAL ALGEBRA

Let $A,B,\dots$ be commutative unitary rings or even integral domain with
fields of quotients $Q(A)=K,Q(B)=L,\dots$ containing a field $k$ as a subring
or subfield, for example a polynomial ring $k[x]$ in many indeterminates with
coefficients in $k$ and the corresponding field $k(x)$ of rational functions.
The ideas that led Erich Kähler to the next definitions in 1930 are of two
kinds ([7]):

$\bullet$ The derivative of a polynomial with respect to any one of the
indeterminates is a polynomial while the derivative of a rational function is
a rational function, a reason sufficient for believing that the concept of
derivation could be useful in algebra.

$\bullet$ The variational and linearization procedures presented in the last
section and used for many applications to physics should be extended to
differential algebra in order to obtain the algebraic counterpart of
definition 3.1.3 and proposition 3.1.5, replacing $X$ by a ring $A$ or a field
$K$.

DEFINITION 3.2.1: A derivation from $A$ to an $A$-module $M$ over $k$ is a map
$\delta:A\rightarrow M$ such that $\delta(a+b)=\delta a+\delta
b,\delta(ab)=a\delta b+b\delta a$ with $\delta{\mid}_{k}=0$ and the set of
such maps is denoted by $der_{k}(A,M)$ with $1\times 1=1\Rightarrow\delta
1=0$. When $M=A$, we simply set $der_{k}(A,A)=der_{k}(A)$.

PROPOSITION 3.2.2: Given any $A$-module $M$ and any derivation $\delta\in
der_{k}(A,M)$, there exists a unique $A$-module denoted by ${\Omega}_{A/k}$,
called module of Kähler differentials of $A$ over $k$, a derivation $d\in
der_{k}(A,{\Omega}_{A/k})$ and a unique morphism $f\in
hom_{A}({\Omega}_{A/k},M)$ such that $\delta=f\circ d$ in the following
commutative diagram:

$\begin{array}[]{rcl}A&\stackrel{{\scriptstyle
d}}{{\longrightarrow}}&{\Omega}_{A/k}\\\ &\delta\searrow&\,\,\downarrow f\\\
&&M\end{array}$

The element $da\in{\Omega}_{A/k}$ is called the differential of $a$ and
$der_{k}(A,M)=hom_{A}({\Omega}_{A/k},M)$.

Proof: Let $F$ be the free $A$-module made by the symbols $da,a\in A$ and let
$N$ be the submodule of $F$ generated by $d\alpha,d(a+b)-da-db,d(ab)-adb-bda$
for $\alpha\in k,a,b\in A$. We set ${\Omega}_{A/k}=F/N$ and the derivation
$d=d_{A/k}:A\rightarrow{\Omega}_{A/k}:a\rightarrow da$ is the universal
derivation allowing to define $f:{\Omega}_{A/k}\rightarrow M$ by $(f\circ
d)(a)=f(da)=\delta a$.
Another way is to take into account the limit procedure that is classically
used in analysis, namely
$(x+h)-x=h\Rightarrow(x+h)^{2}-x^{2}=2xh+h^{2}\Rightarrow dx^{2}=2xdx$ when
$h\mapsto 0$ in order to avoid the square quatity $h^{2}$ and so on. For this,
let us denote by $I$ the kernel of the map $A{\otimes}_{k}A\rightarrow
A:a\otimes b\rightarrow ab$ and define ${\Omega}_{A/k}=I/I^{2}$ while setting
$da=1\otimes a-a\otimes 1,\forall a\in A$. Using the bimodule structure of
${}_{A}A_{A}$ while identifying $a\in A$ with $a\otimes 1\in A{\otimes}_{k}A$,
it follows that $d$ is indeed a derivation from $A$ to the $A$-module
${\Omega}_{A/k}$ as we have successively:

$\begin{array}[]{rcl}adb+bda&=&a(1\otimes b-b\otimes 1)+b(1\otimes a-a\otimes
1)\\\ &=&(a\otimes 1)(1\otimes b-b\otimes 1)+(b\otimes 1)(1\otimes a-a\otimes
1)\\\ &=&(1\otimes a)(1\otimes b-b\otimes 1)+(b\otimes 1)(1\otimes a-a\otimes
1)-(1\otimes a-a\otimes 1)(1\otimes b-b\otimes 1)\\\ &=&(1\otimes ab-ab\otimes
1)\hskip 28.45274ptmod(I^{2})\\\ &=&d(ab)\hskip
28.45274ptmod(I^{2})\end{array}$

and thus:

$\begin{array}[]{rcl}da^{2}=1\otimes a^{2}-a^{2}\otimes 1&=&(1\otimes
a+a\otimes 1)(1\otimes a-a\otimes 1)\\\ &=&(a\otimes 1)(1\otimes a-a\otimes
1)+(1\otimes a)(1\otimes a-a\otimes 1)\\\ &=&2(a\otimes 1)(1\otimes a-a\otimes
1)+(1\otimes a-a\otimes 1)(1\otimes a-a\otimes 1)\\\ &=&2(a\otimes 1)(1\otimes
a-a\otimes 1)\hskip 28.45274ptmod(I^{2})\\\ &=&2ada\hskip
28.45274ptmod(I^{2})\end{array}$

$\Box$

Among the elementary properties of the Kähler differentials, we notice that,
if $f:A\rightarrow B$ is a $k$-algebra homomorphism and $M$ is a $B$-module,
then $M$ becomes a $A$-module under the rule $f(a)=b\Rightarrow
am=f(a)m,\forall a\in A,b\in B,m\in M$ and we have the exact sequence of
$B$-modules:

$0\rightarrow der_{A}(B,M)\rightarrow der_{k}(B,M)\rightarrow der_{k}(A,M)$

because $\delta\in der_{k}(B,M)\Rightarrow\delta\circ f\in der_{k}(A,M)$ and
thus $der_{k}(A,M)\simeq hom_{B}(B{\otimes}_{A}{\Omega}_{A/k},M)$.

The proof of the following two propositions is classical and can be found in
([21], p 387-389):

PROPOSITION 3.2.3: (First fundamental exact sequence) We have the exact
sequence of $B$-modules:

$B{\otimes}_{A}{\Omega}_{A/k}\rightarrow{\Omega}_{B/k}\rightarrow{\Omega}_{B/A}\rightarrow
0$

where the first map is a monomorphism when $f$ is a monomorpism. In
particular, if $k\subset K\subset L$ is a chain of field extensions, then one
has the short exact sequence of vector spaces over $L$:

$0\rightarrow
L{\otimes}_{K}{\Omega}_{K/k}\rightarrow{\Omega}_{L/k}\rightarrow{\Omega}_{L/K}\rightarrow
0$

PROPOSITION 3.2.4: (Second fundamental exact sequence) If we have the short
exact sequence $0\rightarrow\mathfrak{a}\rightarrow A\stackrel{{\scriptstyle
f}}{{\longrightarrow}}B\rightarrow 0$, then we have the exact sequence of
$B$-modules:

$\mathfrak{a}/{\mathfrak{a}}^{2}\rightarrow
B{\otimes}_{A}{\Omega}_{A/k}\rightarrow{\Omega}_{B/k}\rightarrow 0$

EXAMPLE 3.2.5: Let $x,y$ be two indeterminates over the field $k=\mathbb{Q}$
and consider the case $A=k[x,y],B=A/(x^{2},y^{2})$. Then the ideal
$\mathfrak{a}=(x^{2},y^{2})\subset A$ is not prime because
$rad(\mathfrak{a})=\mathfrak{p}=(x,y)\in spec(A)$. The image of $x^{3}$ is
$1\otimes 3x^{2}dx$ in $B{\otimes}_{A}{\Omega}_{A/k}$, that is $3x^{2}\otimes
dx=0$ because $x^{2}\in\mathfrak{a}$. A similar comment applies to $y^{3}$ and
it is easy to see that the kernel of the map
$\mathfrak{a}/{\mathfrak{a}}^{2}\rightarrow B{\otimes}_{A}{\Omega}_{A/k}$ is
of the form $ax^{3}+by^{3},\forall a,b\in A$.

Finally, if $S$ is a multiplicatively closed subset of $A$, we may use the
morphism ${\theta}_{S}:A\rightarrow S^{-1}A$ in Proposition 2.2.3, we shall
study the behaviour of derivations and differentials under localization. As
$\delta\in der_{k}(A,M)$ induces a unique derivation $\delta\in
der_{k}(S^{-1}A,S^{-1}M$ through the known formula $\delta(a/s):s\delta
a-a\delta s)/s^{2}$, it follows that the morphism
$der_{k}(S^{-1}A,S^{-1}M)\rightarrow der_{k}(A,S^{-1}M)$ given by
$\delta\rightarrow\delta\circ{\theta}_{S}$ is an epimorphism. We obtain the
short exact sequence:

$0\rightarrow der_{A}(S^{-1}A,S^{-1}M)\rightarrow
der_{k}(S^{-1}A,S^{-1}M)\rightarrow der_{k}(A,S^{-1}M)\rightarrow 0$

and thus the short exact sequence:

$0\rightarrow
S^{-1}A{\otimes}_{A}{\Omega}_{A/k}\rightarrow{\Omega}_{S^{-1}A/k}\rightarrow{\Omega}_{S^{-1}A/A}\rightarrow
0$

Taking into account the previous standard formula, it follows that
${\Omega}_{S^{-1}A/A}=0$ and we obtain:

PROPOSITION 3.2.6: There is an isomorphism
$S^{-1}A{\otimes}_{A}{\Omega}_{A/k}\simeq{\Omega}_{S^{-1}A/k}$.

EXAMPLE 3.2.7: We now present in an independent manner a few OD or PD cases
showing the difficulties met when studying differential ideals and ask the
reader to revisit them later on while reading the main Theorems. As only a few
results will be proved, the interested reader may look at [18] or [20] for
more details and compare to [9] or [11].

$\bullet\,\,OD\,\,1$: If $k=\mathbb{Q}$, $y$ is a differential indeterminate
and $d_{x}$ is a formal derivation, we may set
$d_{x}y=y_{x},d_{x}y_{x}=y_{xx}$ and so on in order to introduce the
differential ring $A=k[y,y_{x},y_{xx},...]=k\\{y\\}$. We consider the
differential ideal $\mathfrak{a}\subset A$ generated by the differential
polynomial $P=y^{2}_{x}-4y$. We have $d_{x}P=2y_{x}(y_{xx}-2)$ and
$\mathfrak{a}$ cannot be a prime differential ideal, $\dots$ and so on. After
no less than $4$ differentiations, we let the reader discover that
$y_{xxx}^{5}\in\mathfrak{a}\Rightarrow y_{xxx}\in rad(\mathfrak{a})$ and thus
$\mathfrak{a}$ is neither prime nor perfect, that is equal to its radical, but
$rad(\mathfrak{a})$ is perfect as it is the intersection of the prime
differential ideal generated by $y$ with the prime differential ideal
generated by $y_{x}^{2}-4y$ and $y_{xx}-2$, both containing $y_{xxx}$.

$\bullet\,\,OD\,\,2$: With the same notations, let us consider the
differential ideal $\mathfrak{a}\subset A$ generated by the differential
polynomial $P=y^{2}_{x}-4y^{3}$. We have $d_{x}P=2y_{x}(y_{xx}-6y^{2})$ and
$\mathfrak{a}$ cannot be prime differential ideal. Hence, we must have either
$y_{x}=0\Rightarrow y=0$ or $y_{xx}-6y^{2}=0$ and so on. After 3
differentiations we obtain $(y_{xx}-6y^{2})^{4}\in\mathfrak{a}\Rightarrow
y_{xx}-6y^{2}\in rad(\mathfrak{a})$ and thus $\mathfrak{a}$ is neither prime
nor perfect as before but $rad(\mathfrak{a})$ is the prime differential ideal
generated by $y_{x}^{2}-4y^{3}$ and $y_{xx}-6y^{2}$.

$\bullet\,\,PD\,\,1$: If $k=\mathbb{Q}$ as before, $y$ is a differential
indeterminate and $(d_{1},d_{2})$ are two formal derivations, let us consider
the differential ideal generated by $P_{1}=y_{22}-\frac{1}{2}(y_{11})^{2}$ and
$P_{2}=y_{12}-y_{11}$ in $k\\{y\\}$. Using crossed derivatives and
differentiating twice, we get $(y_{111})^{3}\in\mathfrak{a}\Rightarrow
y_{111}\in rad(\mathfrak{a})$ and thus $\mathfrak{a}$ is again neither prime
nor perfect but $rad(\mathfrak{a})$ is a perfect differential ideal and even a
prime differential ideal $\mathfrak{p}$ because we obtain easily from the last
subsection that the resisual differential ring $k\\{y\\}/\mathfrak{p}\simeq
k[y,y_{1},y_{2},y_{11}]$ is a differential integral domain. Its quotient field
is thus the differential field $K=Q(k\\{y\\}/\mathfrak{p})\simeq
k(y,y_{1},y_{2},y_{11})$ with the rules:

$d_{1}y=y_{1},d_{1}y_{1}=y_{11},d_{1}y_{11}=0,d_{2}y=y_{2},d_{2}y_{1}=y_{11},d_{2}y_{11}=0$

as a way to avoid ” looking for solutions ”. The formal linearization is the
linear system $R_{2}\subset J_{2}(E)$ obtained in the last section where it
was defined over ${\cal{R}}_{2}$, but not over $K$, by the two linear second
order PDE:

${\eta}_{22}-y_{11}{\eta}_{11}=0,\hskip
28.45274pt{\eta}_{12}-{\eta}_{11}=0\,\,\,\Rightarrow\,\,\,(y_{11}-1){\eta}_{111}=0$

changing slightly the notations with $\eta=\delta y$ and keeping the letter
$v$ only when looking at the symbols. It is at this point that the problem
starts because ${\cal{R}}_{2}$ is indeed a fibered manifold with arbitrary
parametric jets $(y,y_{1},y_{2},y_{11})$ but
${\cal{R}}_{3}={\rho}_{1}({\cal{R}}_{2})$ is no longer a fibered manifold
because the dimension of its symbol changes when $y_{11}=1$. We understand
therefore that there should be a close link existing between formal
integrability and the search for prime differential ideals or differential
fields. The solution of this problem has been provided as early as in 1983 for
studying the ”Differential Galois Theory ” in ([18]). The idea is to add the
third order PDE $y_{111}=0$ and thus consider the linearized PDE
${\eta}_{111}=0$ obtaining therefore a third order involutive system well
defined over $K$ with symbol $g_{3}=0$.

$\bullet\,\,PD\,\,2$: With the same notations, let us consider the
differential ideal generated by the differential polynomials
$P_{1}=y_{22}-\frac{1}{3}(y_{11})^{3}$ and
$P_{2}=y_{12}-\frac{1}{2}(y_{11})^{2}$ in $k\\{y\\}$. We get:

$P_{1},P_{2}\in\mathfrak{a}\Rightarrow
d_{2}P_{2}-d_{1}P_{1}+y_{11}d_{1}P_{2}=0\Rightarrow{\cal{R}}_{2}\hskip
8.53581ptinvolutive$

with $dim(g_{q})=1,\forall q\geq 1$. As the symbol $g_{2}$ is involutive,
there is an infinite number of parametric jets
$(y,y_{1},y_{2},y_{11},y_{111},...)$ and thus $k\\{y\\}/\mathfrak{a}\simeq
k[y,y_{1},y_{2},y_{11},y_{111},...]$ is a differential integral domain with
$d_{2}y_{2}=y_{22}=\frac{1}{3}(y_{11})^{3},d_{2}y_{11}=y_{112}=y_{11}y_{111},...$.
It follows that $\mathfrak{a}=\mathfrak{p}$ is a prime differential ideal with
$rad(\mathfrak{p})=\mathfrak{p}$. The second order linearized system is:

${\eta}_{22}-(y_{11})^{2}{\eta}_{11}=0,\hskip
28.45274pt{\eta}_{12}-y_{11}{\eta}_{11}=0$

is now well defined over the differential field $K=Q(k\\{y\\}/\mathfrak{p})$
and is involutive.

DEFINITION 3.2.8: A differential ring is a ring $A$ with a finite number of
commuting derivations $({\partial}_{1},...,{\partial}_{n})$ that can be
extended to derivations of the ring of quotients $Q(A)$ as we already saw. We
shall suppose from now on that $A$ is even an integral domain and introduce
the differential field $K=Q(A)$. For example, if $x^{1},...,x^{n}$ are
indeterminates over $\mathbb{Q}$, then
$\mathbb{Q}[x]=\mathbb{Q}[x^{1},...,x^{n}]$ is a differential ring with
quotient differential field $\mathbb{Q}(x)$.

If $K$ is a differential field as above and $(y^{1},...,y^{m})$ are
indeterminates over $K$, we transform the polynomial ring
$K\\{y\\}={lim}_{q\rightarrow\infty}K[y_{q}]$ into a differential ring by
introducing as usual the formal derivations
$d_{i}={\partial}_{i}+y^{k}_{\mu+1_{i}}\partial/\partial y^{k}_{\mu}$ and we
shall set $K<y>=Q(K\\{y\\})$.

DEFINITION 3.2.9: We say that $\mathfrak{a}\subset K\\{y\\}$ is a differential
ideal if it is stable by the $d_{i}$, that is if
$d_{i}a\in\mathfrak{a},\forall a\in\mathfrak{a},\forall i=1,...,n$. We shall
also introduce the radical $rad(\mathfrak{a})=\\{a\in A\mid\exists
r,a^{r}\in\mathfrak{a}\\}\supseteq\mathfrak{a}$ and say that $\mathfrak{a}$ is
a perfect (or radical) differential ideal if $rad(\mathfrak{a})=\mathfrak{a}$.
If $S$ is any subset of $A$, we shall denote by $\\{S\\}$ the differential
ideal generated by $S$ and introduce the (non-differential) ideal
${\rho}_{r}(S)=\\{d_{\nu}a\mid a\in S,0\leq\mid\nu\mid\leq r\\}$ in $A$.

LEMMA 3.2.10: If $\mathfrak{a}\subset A$ is differential ideal, then
$rad(\mathfrak{a})$ is a differential ideal containing $\mathfrak{a}$.

Proof: If $d$ is one of the derivations, we have
$a^{r-1}da=\frac{1}{r}da^{r}\in\\{a^{r}\\}$ and thus:

$(r-1)a^{r-2}(da)^{2}+a^{r-1}d^{2}a\in\\{a^{r}\\}\Rightarrow
a^{r-2}(da)^{3}\in\\{a^{r}\\},...\Rightarrow(da)^{2r-1}\in\\{a^{r}\\}$

$\Box$

LEMMA 3.2.11: If $\mathfrak{a}\subset K\\{y\\}$, we set
${\mathfrak{a}}_{q}=\mathfrak{a}\cap K[y_{q}]$ with
${\mathfrak{a}}_{0}=\mathfrak{a}\cap K[y]$ and
${\mathfrak{a}}_{\infty}=\mathfrak{a}$. We have in general
${\rho}_{r}({\mathfrak{a}}_{q})\subseteq{\mathfrak{a}}_{q+r}$ and the problem
will be to know when we may have equality.

We shall say that a differential extension $L=Q(K\\{y\\}/\mathfrak{p})$ is a
finitely generated differential extension of $K$ and we may define the
evaluation epimorphism $K\\{y\\}\rightarrow K\\{\eta\\}\subset L$ with kernel
$\mathfrak{p}$ by calling $\eta$ or $\bar{y}$ the residue of $y$ modulo
$\mathfrak{p}$. If we study such a differential extension $L/K$, by analogy
with Section 2, we shall say that $R_{q}$ or $g_{q}$ is a vector bundle over
${\cal{R}}_{q}$ if one can find a certain number of maximum rank determinant
$D_{\alpha}$ that cannot be all zero at a generic solution of
${\mathfrak{p}}_{q}$ defined by differential polynomials $P_{\tau}$, that is
to say, according to the Hilbert Theorem of Zeros, we may find polynomials
$A_{\alpha},B_{\tau}\in K\\{y_{q}\\}$ such that
${\sum}_{\alpha}A_{\alpha}D_{\alpha}+{\sum}_{\tau}B_{\tau}P_{\tau}=1$. The
following Lemma will be used in the next important Theorem:

LEMMA 3.2.12: If $\mathfrak{p}$ is a prime differential ideal of $K\\{y\\}$,
then, for $q$ sufficiently large, there is a polynomial $D\in K[y_{q}]$ such
that $D\notin{\mathfrak{p}}_{q}$ and :

$D{\mathfrak{p}}_{q+r}\subset
rad({\rho}_{r}({\mathfrak{p}}_{q}))\subset{\mathfrak{p}}_{q+r},\hskip
28.45274pt\forall r\geq 0$

THEOREM 3.2.13: (Primality test) Let ${\mathfrak{p}}_{q}\subset K[y_{q}]$ and
${\mathfrak{p}}_{q+1}\subset K[y_{q+1}]$ be prime ideals such that
${\mathfrak{p}}_{q+1}={\rho}_{1}({\mathfrak{p}}_{q})$ and
${\mathfrak{p}}_{q+1}\cap K[y_{q}]={\mathfrak{p}}_{q}$. If the symbol $g_{q}$
of the algebraic variety ${\cal{R}}_{q}$ defined by ${\mathfrak{p}}_{q}$ is
$2$-acyclic and if its first prolongation $g_{q+1}$ is a vector bundle over
${\cal{R}}_{q}$, then $\mathfrak{p}={\rho}_{\infty}({\mathfrak{p}}_{q})$ is a
prime differential ideal with $\mathfrak{p}\cap
K[y_{q+r}]={\rho}_{r}({\mathfrak{p}}_{q}),\forall r\geq 0$.

COROLLARY 3.2.14: Every perfect differential ideal of $\\{y\\}$ can be
expressed in a unique way as the non-redundant intersection of a finite number
of prime differential ideals.

COROLLARY 3.2.15: (Differential basis) If $\mathfrak{r}$ is a perfect
differential ideal of $K\\{y\\}$, then we have
$\mathfrak{r}=rad({\rho}_{\infty}({\mathfrak{r}}_{q}))$ for $q$ sufficiently
large.

EXAMPLE 3.2.16: As $K\\{y\\}$ is a polynomial ring with an infinite number of
variables it is not noetherian and an ideal may not have a finite basis. With
$K=\mathbb{Q},n=1$ and $d=d_{x}$, then
$\mathfrak{a}=\\{yy_{x},y_{x}y_{xx},y_{xx}y_{xxx},...\\}\Rightarrow(y_{x})^{2}+yy_{xx}\in\mathfrak{a}\Rightarrow
rad(\mathfrak{a})=\\{y_{x}\\}$ is a prime differential ideal.

PROPOSITION 3.2.17: If $\zeta$ is differentially algebraic over $K<\eta>$ and
$\eta$ is differentially algebraic over $K$, then $\zeta$ is differentially
algebraic over $K$. Setting $\xi=\zeta-\eta$, it follows that, if $L/K$ is a
differential extension and $\xi,\eta\in L$ are both differentially algebraic
over $K$, then $\xi+\eta$, $\xi\eta$ and $d_{i}\xi$ are differentially
algebraic over $K$.

If $L=Q(K\\{y\\}/\mathfrak{p})$, $M=Q(K\\{z\\}/\mathfrak{q})$ and
$N=Q(K\\{y,z\\}/\mathfrak{r})$ are such that $\mathfrak{p}=\mathfrak{r}\cap
K\\{y\\}$ and $\mathfrak{q}=\mathfrak{r}\cap K\\{z\\}$, we have the two towers
$K\subset L\subset N$ and $K\subset M\subset N$ of differential extensions and
we may therefore define the new tower $K\subseteq L\cap
M\subseteq<L,M>\subseteq N$. However, if only $L/K$ and $M/K$ are known and we
look for such an $N$ containing both $L$ and $M$, we may use the universal
property of tensor products an deduce the existence of a differential morphism
$L{\otimes}_{K}M\rightarrow N$ by setting $d(a\otimes b)=(d_{L}a)\otimes
b+a\otimes(d_{M}b)$ whenever $d_{L}\mid K=d_{M}\mid K=\partial$. Looking for
an abstract composite differential field amounts therefore to look for a prime
differential ideal in $L{\otimes}_{K}M$ which is a direct sum of integral
domains (See [18] for more details).

DEFINITION 3.2.18: A differential extension $L$ of a differential field $K$ is
said to be differentially algebraic over $K$ if every element of $L$ is
differentially algebraic over $K$. The set of such elements is an intermediate
differential field $K^{\prime}\subseteq L$, called the differential algebraic
closure of $K$ in $L$. If $L/K$ is a differential extension, one can always
find a maximal subset $S$ of elements of $L$ that are differentially
transcendental over $K$ and such that $L$ is differentially algebraic over
$K<S>$. Such a set is called a differential transcendence basis and the number
of elements of $S$ is called the differential transcendence degree of $L/K$.

THEOREM 3.2.19: If $L/K$ is a finitely generated differential extension, then
any intermediate differential field $K^{\prime}$ between $K$ and $L$ is also
finitely generated over $K$.

THEOREM 3.2.20: The number of elements in a differential basis of $L/K$ does
not depent on the generators of $L/K$ and his value is $difftrd(L/K)=\alpha$.
Moreover, if $K\subset L\subset M$ are differential fields, then
$difftrd(M/K)=difftrd(M/L)+difftrd(L/K)$.

Comparing the differential geometric approach to nonlinear algebraic systems
with the differential algebraic approach just presented while setting $\delta
y_{q}={\eta}_{q}$, we obtain:

COROLLARY 3.2.21: When $L/K$ is a finitely generated differential extension,
then ${\Omega}_{L/K}$ is a differential module over the differential ring
$L{\otimes}_{K}K[d]=L[d]$ of differential operators with coefficients in $L$.
The linearized ”system ” $R=hom_{L}({\Omega}_{L/K},L)$ is thus a (left)
differential module for the Spencer operator like in the linear framework.

It is not evident to grasp these results in order to apply them to control
theory or mathematical physics for two reasons. The first is that the formal
theory of nonlinear systems has not been accepted by differential geometers
because of the homological background based on the so-called ”vertical
machinery” and the systematic use of the Spencer $\delta$-cohomology. The
recent study of the Schwarzschild and Kerr metrics (Compare [35] to [1]) is
providing a good example of such a poor situation. The second is the fact
that, when $K$ is a true differential field and $M$ is differential module
defined over the noncommutative ring $K[d_{1},\dots,d_{n}]=K[d]$ of
differential operators with coefficients in $K$, then the ”system”
$R=hom_{K}(M,K)$ is still not used today because its differential structure
highly depends on the Spencer operator which has never been introduced in
physics. As a good example, we may quote the fact that the Cosserat couple
stress equations are just described by the formal adjoint of the linear
Spencer operator ([19]).

3.3) NONLINEAR CONTROL THEORY

As we have already explained in ([18]), the generalized ” Bäcklund problem ”
is nothing else than the study of nonlinear differential correspondences in
the theory of differential elimination. We shall provide, below and
successively, a differential geometric definition followed by a differential
algebraic definition and all the problem will be to establish a link between
them.

When $X$ is a manifold of dimension $n$, let us consider two fibered manifolds
over $X$, namely ${\cal{E}}$ with lcal coordinates $(x,y)$ and ${\cal{F}}$
with local coordinates $(x,z)$. The fibered roduct
${\cal{E}}{\times}_{X}{\cal{F}}$ is a fibered manifold over $X$ with ocal
coordinates $(x,y,z)$ and we have the canonical identification:

$J_{q}({\cal{E}}{\times}_{X}{\cal{F}})=J_{q}({\cal{E}}){\times}_{X}J_{q}({\cal{F}})$

with local coordinates $(x,y_{q},z_{q})$.
For most applications, we shall suppose that ${\cal{E}}=X\times Y$ and
${\cal{F}}=X\times Z$.

DEFINITION 3.3.1: Let ${\cal{R}}_{q}\subset
J_{q}({\cal{E}}{\times}_{X}{\cal{F}})$ be a nonlinear system of order $q$ on
${\cal{E}}{\times}_{X}{\cal{F}}$ called a differential correspondence between
$(y,z)$. When $r\rightarrow\infty$, we may consider the resolvent systems
${\cal{P}}_{q+r}\subset J_{q+r}({\cal{E}})$ for $y$ and
${\cal{Q}}_{q+r}\subset J_{q+r}({\cal{F}})$ for $z$, induced by the canonical
projections of ${\cal{E}}{\times}_{X}{\cal{F}}$ onto ${\cal{E}}$ and
${\cal{F}}$ respectively.

Roughly, finding ${\cal{P}}$ amounts to eliminate $z$ while finding
${\cal{Q}}$ amounts to eliminate $y$ and we shall only consider te first
problem as the second will be similar.

$\bullet$ In the linear case, pushing $y$ on the left and $z$ on the right, we
are left with the search of the CC for $y$ or the CC for $z$ that may be quite
difficult. One of the best examples has been provided by M. Janet with the
second order system (See [20] or [22] for details):

$y_{33}-x^{2}y_{11}=z^{1},\hskip 56.9055pty_{22}=z^{2}$

where $y=f(x)$ can be given arbitrarily for getting $z=g(x)$ while $z=g(x)$
must satisfy one CC of order $3$ and one CC of order $6$.

$\bullet$ In the nonlinear case, we have ([18],[21]):

THEOREM OF THE RESOLVENT SYSTEMS 3.3.2: In general, one may find two integers
$r,s\geq 0$ such that ${\cal{R}}^{(s)}_{q+r}$ is formally integrable
(involutive) with formally integrable (involutive) projections
${\cal{P}}^{(s)}_{q+r}\subset J_{q+r}({\cal{E}})$ and
${\cal{Q}}^{(s)}_{q+r}\subset J_{q+r}({\cal{F}})$. Moreover, $r$ and $s$ can
be (tentatively) found by a finite algorithm preserving the symmetry existing
between $\cal{E}$ and $\cal{F}$.

Proof: First of all, we know that, in general, one can find the two integers
$r,s\geq 0$ in such a way that ${\cal{R}}^{(s)}_{q+r}$ is formally integrable
(involutive). Hence, using the commutative and exact diagram:

$\begin{array}[]{cccl}{\cal{R}}_{q+r+s}&\longrightarrow&{\cal{P}}_{q+r+s}&\rightarrow
0\\\ \downarrow&&\downarrow&\\\
{\cal{R}}^{(s)}_{q+r}&\longrightarrow&{\cal{P}}^{(s)}_{q+r}&\rightarrow 0\\\
\cap&&\cap&\\\ {\cal{R}}_{q+r}&\longrightarrow&{\cal{P}}_{q+r}&\rightarrow
0\end{array}$

we may suppose, without any loss of generality, that ${\cal{R}}_{q}$ is
formally integrable (involutive).
Now, chasing in the commutative diagram:

$\begin{array}[]{ccccccl}&&J_{s}({\cal{R}}_{q+r})&&\longrightarrow&&J_{s}(J_{q+r}({\cal{E}}{\times}_{X}{\cal{F}})\\\
&\nearrow&|&&&\nearrow&\hskip 22.76219pt|\\\
{\cal{R}}_{q+r+s}&&\longrightarrow&&J_{q+r+s}({\cal{E}}{\times}_{X}{\cal{F}})&&\hskip
22.76219pt|\\\ |&&\downarrow&&|&&\hskip 22.76219pt\downarrow\\\
|&&J_{s}({\cal{P}}_{q+r})&&\longrightarrow&&J_{s}(J_{q+r}({\cal{E}}))\\\
\downarrow&\nearrow&&&\downarrow&\nearrow&\\\
{\cal{P}}_{q+r+s}&&\longrightarrow&&J_{q+r+s}({\cal{E}})&&\end{array}$

we obtain therefore
${\cal{P}}_{q+r+s}\subseteq{\rho}_{s}({\cal{P}}_{q+r})=J_{s}({\cal{P}}_{q+r})\cap
J_{q+r+s}({\cal{E}})\subset J_{s}(J_{q+r}({\cal{E}})),\forall r,s\geq 0$.
Then, chasing in the commutative diagram:

$\begin{array}[]{cccl}{\cal{R}}_{q+r+s}&\longrightarrow&{\cal{P}}_{q+r+s}&\rightarrow
0\\\ \downarrow&&\downarrow&\\\
{\cal{R}}_{q+r}&\longrightarrow&{\cal{P}}_{q+r}&\rightarrow 0\\\
\downarrow&&&\\\ 0&&&\end{array}$

we notice that
${\pi}^{q+r+s}_{q+r}:{\cal{P}}_{q+r+s}\longrightarrow{\cal{P}}_{q+r}$ is an
epimorphism $\forall r,s\geq 0$.
Finally, chasing in the commutative and exact diagram:

$\begin{array}[]{rcccl}0\rightarrow&{\cal{P}}_{q+r+s}&\longrightarrow&{\rho}_{s}({\cal{P}}_{q+r})&\rightarrow
0\\\ &\downarrow&&\downarrow&\\\
0\rightarrow&{\cal{P}}_{q+r}&=&{\cal{P}}_{q+r}&\rightarrow 0\\\
&\downarrow&&&\\\ &0&&&\end{array}$

we deduce that each ${\cal{P}}_{q+r}$ is formaly integrable at each
$q+r,\forall r\geq 0$, though not always formally integrable as we shall see
on examples.
Looking at the symbol $h$ of ${\cal{P}}$, we have
$h_{q+r+s}\subseteq{\rho}_{s}(h_{q+r})$ over ${\cal{P}}_{q+r+s}$. According to
standard Noetherian arguments, such a situation is stabilizing for $r$ and $s$
large enough but such an approach is not constructive in general.

For this reason, we shall prefer to use a different approach which is closer
to the one met in the case of linear differential correspondences. For this,
if $z=g(x)$ is an arbitrary section of ${\cal{F}}$, we shall consider the new
system for $y$ defined by ${\cal{A}}_{q}=j_{q}(f)^{-1}({\cal{R}}_{q})$ over
$K<g>$. Such a system, which is in general neither involutive nor even
formally integrable as we shall see on examples, may also be not even
compatible as it may not provide a fibered manifold but this way may give
informations on the order of the OD or PD equations that should be satisfied
by $z$. A similar procedure could be used by setting $y=f(x)$ and introducing
${\cal{B}}_{q}=j_{q}(f)^{-1}({\cal{R}}_{q})$ in order to obtain a system for
$z$ over $K<f>$.
$\Box$

Let us now turn to the differential algebraic counterpart.

DEFINITION 3.3.3: If $K$ is a differential field and we have a differential
algebraic correspondence defined by a prime differential ideal
$\mathfrak{r}\subset K\\{y,z\\}$, we may define the resolvent system for $y$
by the resolvent differential ideal $\mathfrak{p}=\mathfrak{r}\cap K\\{y\\}$
and the resolvent system for $z$ by the resolvent differential ideal
$\mathfrak{q}=\mathfrak{r}\cap K\\{z\\}$.

LEMMA 3.3.4: The resolvent ideal for $y$ is the prime differential resolvent
ideal $\mathfrak{p}=\mathfrak{r}\cap K\\{y\\}$ for which one can find a
differential basis. Similarly, the prime differential resolvent ideal for $z$
is $\mathfrak{q}=\mathfrak{r}\cap K\\{z\\}$.

Proof: We have the commutative and exact diagram:

$\begin{array}[]{rcccccl}&0&&0&&0&\\\ &\downarrow&&\downarrow&&\downarrow&\\\
0\rightarrow&\mathfrak{p}&\longrightarrow&K\\{y\\}&\longrightarrow&A&\rightarrow
0\\\ &\downarrow&&\downarrow&&\downarrow&\\\
0\rightarrow&\mathfrak{r}&\longrightarrow&K\\{y,z\\}&\longrightarrow&B&\rightarrow
0\end{array}$

First of all, $B$ is an integral domain because $\mathfrak{r}$ is a prime
differential ideal. It follows from a chase that the induced morphism
$A\rightarrow B$ is a monomorphism and $A\simeq im(A)\subset B$ is thus also
an integral domain, a result showing that $\mathfrak{p}$ is a prime
differential ideal. It is essential to notice that projections of ideals
cannot be used in the nonlinear framework. Hence, the idea is to reduce the
study of differential algebraic correspondences to the study of purely
algebraic correspondences.
$\Box$

We end this last section with a few basic motivating examples showing the
importance of the non-commutative localization of integral domains for
explicit computations and applications. We hope therefore that these examples
could be used as test examples for future applications of computer algebra
(Compare to [16]).

EXAMPLE 3.3.5: With $m=1,n=2,q=2,K=\mathbb{Q}(x^{1},x^{2})$ while using local
coordinates $(x^{1},x^{2},y)$ for the fibered manifold ${\cal{E}}$ let us
consider anew the nice example presented by J. Johnson in ([8]), namely the
nonlinear system ${\cal{R}}_{2}\in J_{2}({\cal{E}})$ defined by the two
algebraic PD equations :

$P_{1}\equiv y_{22}-x^{2}y=0,\,\,\,\,P_{2}\equiv y_{12}-(y)^{2}=0$

We let the reader prove successively as an exercise that:
${\cal{R}}^{(1)}_{2}$ is adding $2yy_{2}-x^{2}y_{1}=0$.
${\cal{R}}^{(2)}_{2}$ is adding $2(y_{2})^{2}-y_{1}+x^{2}(y)^{2}=0$.
${\cal{R}}^{(3)}_{2}$ is adding $6x^{2}yy_{2}=0$ and thus $y_{1}=0$.
${\cal{R}}^{(4)}_{2}$ is adding $(y)^{2}=0$.
Accordingly, the prime ideal ${\mathfrak{p}}_{2}\in
K[y,y_{1},y_{2},y_{11},y_{12},y_{22}]$ generated by the two given differential
polynomias $(P_{1},P_{2})$ is such that $y\in
rad({\rho}_{4}({\mathfrak{p}}_{2}))\Rightarrow
rad({\rho}_{\infty}({\mathfrak{p}}_{2}))=(y)$, a result not evident at first
sight and leading to the trivial differential extension $L=K$. The
linearization procedure is even less evident. Indeed, starting with the
linearized second order system:

$\delta P_{1}\equiv{\eta}_{22}-x^{2}\eta=0,\,\,\,\delta
P_{2}\equiv{\eta}_{12}-2y\eta=0$

we let the reader prove that we successively get:
$R^{(1)}_{2}$ is adding $2y{\eta}_{2}-x^{2}{\eta}_{1}+2y_{2}\eta=0$.
$R^{(2)}_{2}$ is adding $4y_{2}{\eta}_{2}-{\eta}_{1}+2x^{2}y\eta=0$.
$R^{(3)}_{2}$ is adding ${\eta}_{1}=0$.
$R^{(4)}_{2}$ is adding $y\eta=0$ but one cannot conclude.
Such an example is proving that, in general, one must start from a formally
integrable or even involutive system in order to be able to define the module
of Kähler differentials for the differential extension $L/K$.

EXAMPLE 3.3.6: (Burgers) With $n=2,m=2$, local coordinates $(x^{1},x^{2},y,z)$
and differential field $K=\mathbb{Q}$, let us consider the algebraic first
order involutive system ${\cal{R}}_{1}$ defined by two differential algebraic
PD equations:

$\left\\{\begin{array}[]{l}z_{2}-y=0\\\
y_{2}-z_{1}+(y)^{2}=0\end{array}\right.\framebox{ $\begin{array}[]{cc}1&2\\\
1&2\end{array}$ }$

These two differential polynomials generate a prime differential ideal
$\mathfrak{r}\subset K\\{y,z\\}$ and provide thus a differential extension
$N/K$. Indeed, $K[y,y_{1},y_{2};z,z_{1},z_{2}]/{\mathfrak{r}}_{1}\simeq
K[y,y_{1};z,z_{1}]$ is an integral domain and ${\mathfrak{r}}_{1}$ is a prime
ideal. Then, using one prolongation, we may introduce the following second
order system ${\cal{R}}_{2}={\rho}_{1}({\cal{R}}_{1})$:

$\left\\{\begin{array}[]{l}z_{22}-y_{2}=0\\\ y_{22}+2yy_{2}-y_{1}=0\\\
z_{12}-y_{1}=0\\\ y_{12}-z_{11}+2yy_{1}=0\\\ z_{2}-y=0\\\
y_{2}-z_{1}+(y)^{2}=0\end{array}\right.\framebox{$\begin{array}[]{cc}1&2\\\
1&2\\\ 1&\bullet\\\ 1&\bullet\\\ \bullet&\bullet\\\
\bullet&\bullet\end{array}$ }$

and use the Janet tabular to prove that it is a nonlinear involutive system.
It follows that $K[y,...,y_{22};z,...,z_{22}]/{\mathfrak{r}}_{2}\simeq
K[y,y_{1},y_{11};z,z_{1},z_{11}]$ is an integral domain and
${\mathfrak{r}}_{2}$ is a prime ideal. Thanks to Theorem $3.2.13$, we obtain
thus finally $K\\{y,z\\}/\mathfrak{r}\simeq
K[y,y_{1},y_{11},...;z,z_{1},z_{11},...]$ which is also an integral domain. It
follows hat $\mathfrak{p}=\mathfrak{r}\cap K\\{y\\}$ and
$\mathfrak{q}=\mathfrak{r}\cap K\\{z\\}$ are prime differetial ideals.
Taking now any section $y=f(x)$, we obtain the system
${\cal{B}}_{1}={j_{1}(f)}^{-1}({\cal{R}}_{1})$ for $z$:

$\left\\{\begin{array}[]{l}z_{2}-f(x)=0\\\
z_{1}-{\partial}_{2}f(x)-(f(x))^{2}=0\end{array}\right.$

and its first prolongation ${\cal{B}}_{2}={j_{2}(f)}^{-1}({\cal{R}}_{2})$ for
$z$:

$\left\\{\begin{array}[]{l}z_{22}-{\partial}_{2}f(x)=0\\\
z_{12}-{\partial}_{1}f(x)=0\\\
z_{11}-{\partial}_{12}f(x)-2f(x){\partial}_{1}f(x)=0\\\ z_{2}-f(x)=0\\\
z_{1}-{\partial}_{2}f(x)-(f(x))^{2}=0\\\
{\partial}_{22}f(x)+2f(x){\partial}_{2}f(x)-{\partial}_{1}f(x)=0\end{array}\right.$

First of all, this is a fibered manifold if and only if $f$ is solution of the
second order system ${\cal{P}}_{2}$ defined by the single second order PD
equation:

$y_{22}+2yy_{2}-y_{1}=0$

which is the resolvent system for $y$ generating the prime differential ideal
$\mathfrak{p}\subset K\\{y\\}$ allowing to define a differential extension
$L=Q(K\\{y\\}/\mathfrak{p})$ of $K$ and we have $\mathfrak{p}=\mathfrak{r}\cap
K\\{y\\}\Rightarrow L\subset N$.
We are thus left with the first order (nonlinear) system for $z$:

$\left\\{\begin{array}[]{l}z_{2}-f(x)=0\\\
z_{1}-{\partial}_{2}f(x)-(f(x))^{2}=0\end{array}\right.\framebox{
$\begin{array}[]{cc}1&2\\\ 1&2\end{array}$ }$

which is easily seen to be involutive for any solution $y=f(x)$ of
${\cal{P}}_{2}$.
Taking finally any section $z=g(x)$, we obtain the system
${\cal{A}}_{1}=j_{1}(g)^{-1}({\cal{R}}_{1})$:

$\left\\{\begin{array}[]{l}y_{2}+(y)^{2}-{\partial}_{1}g(x)=0\\\
y-{\partial}_{2}g(x)=0\end{array}\right.$

and the projecion ${\cal{A}}^{(1)}_{1}$ of its first prolongation
${\cal{A}}_{2}={j_{2}(g)}^{-1}({\cal{R}}_{2})$:

$\left\\{\begin{array}[]{l}y_{2}+(y)^{2}-{\partial}_{1}g=0\\\
y_{1}-{\partial}_{12}g=0\\\ y-{\partial}_{2}g=0\\\
{\partial}_{22}g(x)+({\partial}_{2}g(x))^{2}-{\partial}_{1}g(x)=0\end{array}\right.$

is compatible if and only if $g$ is solution of the second order system
${\cal{Q}}_{2}$ defined by the single second order PD equation obtained after
substitution of $y={\partial}_{2}g(x)$:

$z_{22}+(z_{2})^{2}-z_{1}=0$

which is the resolvent system for $z$ generating the prime differential ideal
$\mathfrak{q}=\mathfrak{r}\cap K\\{z\\}$ allowing to define a differential
extension $M=Q(K\\{z\\}/\mathfrak{q})$ of $K$ and we have
$\mathfrak{q}=\mathfrak{r}\cap K\\{z\\}\Rightarrow M\subset N$.
We are thus left with the only zero order (linear) equation for $y$, namely:

$y-{\partial}_{2}g(x)=0$

for any solution $z=g(x)$ of ${\cal{Q}}_{2}$. The differential correspondence
that must be used is thus ${\cal{R}}_{2}$.
Both $L,M$ and $N$ are differential algebraic extensions of $K$ of zero
differential transcendence degree.

EXAMPLE 3.3.7: (Korteweg-de Vries) With the same notations, we let the reader
provide the details of the following similar example with the second order
nonliear differential correspondence ${\cal{R}}_{2}$:

$z_{2}+(z)^{2}+2y=0,\hskip
14.22636pty_{22}+2y(z)^{2}+4(y)^{2}-2zy_{2}-\frac{1}{2}z_{1}=0$

by exhibiting the nonlinear formally integrable involutive system
${\cal{R}}^{(1)}_{2}$.

$\left\\{\begin{array}[]{l}z_{22}+2zz_{2}+2y_{2}=0\\\
y_{22}+2y(z)^{2}+4(y)^{2}-2zy_{2}-\frac{1}{2}z_{1}=0\\\
z_{12}+2zz_{1}++2y_{1}=0\\\ z_{2}+(z)^{2}+2y=0\end{array}\right.\framebox{
$\begin{array}[]{cc}1&2\\\ 1&2\\\ 1&\bullet\\\ \bullet&\bullet\end{array}$ }$

for $(y,z)$ such that ${\cal{R}}^{(1)}_{3}={\rho}_{1}({\cal{R}}^{(1)}_{2})$
according to Theorem $3.1.13$, with characters
${\alpha}^{1}_{2}=3,{\alpha}^{2}_{2}=0$ (Compare to [18]).
Taking now any section $y=f(x)$, we obtain the system
${\cal{B}}_{2}=j_{2}(f)^{-1}({\cal{R}}^{(1)}_{2})$ for $z$ which is the first
prolongation of the first order (nonlinear) system ${\cal{B}}_{1}$ defined by:

$\left\\{\begin{array}[]{l}z_{2}+(z)^{2}+2f(x)=0\\\
z_{1}+4{\partial}_{2}f(x)z-4f(x)(z)^{2}-8(f(x))^{2}-2{\partial}_{22}f(x)=0\end{array}\right.\framebox{
$\begin{array}[]{cc}1&2\\\ 1&\bullet\end{array}$ }$

Using crossed derivatives and tedious but elementary substitutions, this
system is involutive if and only if $y=f(x)$ is a solution of the third order
involutive resolvent system ${\cal{P}}^{(1)}_{3}$ for $y$:

$y_{222}+y_{1}+12yy_{2}=0$

Similarly, taking now any section $z=g(x)$, we obtain the system
${\cal{A}}_{2}=j_{2}(g)^{-1}({\cal{R}}^{(1)}_{2})$ defined by:

$\left\\{\begin{array}[]{l}2y_{22}+4(g(x))^{2}+8(y)^{2}-4g(x)y_{2}-{\partial}_{1}g(x)=0\\\
2y_{2}+{\partial}_{22}g(x)+2g(x){\partial}_{2}g(x)=0\\\
2y_{1}+{\partial}_{12}g(x)+2g(x){\partial}_{1}g(x)=0\\\
2y+{\partial}_{2}g(x)+(g(x))^{2}=0\end{array}\right.$

Differentiating the second equation with respect to $x^{2}$ and substracting
the first while using the other equations, we discover that this system is
compatible if and only if $z=g(x)$ is a solution of the third order involutive
resolvent system ${\cal{Q}}^{(1)}_{3}$ for $z$:

$z_{222}+z_{1}-6(z)^{2}z_{2}=0$

It follows that we are left with a single zero order equation for $y$, namely:

$2y+{\partial}_{2}g(x)+(g(x))^{2}=0$

for any solution $z=g(x)$ of ${\cal{Q}}_{2}$. The differential correspondence
that must be used is thus ${\cal{R}}^{(1)}_{3}$.

EXAMPLE 3.3.8: With $n=1,m=2,K=\mathbb{Q}$, let us consider the single
input/single output (SISO) nonlinear control system $P\equiv
y^{1}y^{2}_{x}-y^{1}_{x}-a=0$ with a constant parameter $a\in K$. The
differential ideal $\mathfrak{p}\subset K\\{y^{1},y^{2}\\}=K\\{y\\}$ generated
by $P$ is prime because $K\\{y\\}/\mathfrak{p}=K[y^{1};y^{2},y^{2}_{x},...]$
is an integral domain and we set as usual $L=Q(K\\{y\\}/\mathfrak{p})$. The
corresponding linearized system is
$y^{1}{\eta}^{2}_{x}+y^{2}_{x}{\eta}^{1}-{\eta}^{1}_{x}=0$. Multiplying by a
test function $\lambda$ and integrating by parts, the adjoint operator is:

$\left\\{\begin{array}[]{rcl}{\eta}^{1}&\rightarrow&d\lambda+y^{2}_{x}\lambda={\mu}^{1}\\\
{\eta}^{2}&\rightarrow&-y^{1}d\lambda-y^{1}_{x}\lambda={\mu}^{2}\end{array}\right.$

Multiplying the first OD equation by $y^{1}$, the second by $1$ and adding
them, we get $a\lambda=y^{1}{\mu}^{1}+{\mu}^{2}$. As $L[d]$ is a principal
ideal domain, it follows that $M={\Omega}_{L/K}$ is a torsion-free and thus
free differential module over $L[d]$ if and only if this operator is injective
([21],[22]), that is to say if and only if $a\neq 0$.
If $a=0$, then $t(M)$ is generated by
$\omega=y^{1}{\eta}^{2}-{\eta}^{1}=y^{1}\delta y^{2}-\delta y^{1}$ satisfying:

$d\omega=y^{1}\delta y^{2}_{x}+y^{1}_{x}\delta y^{2}-\delta
y^{1}_{x}=\delta(y^{1}y^{2}_{x})+y^{2}_{x}\omega-\delta
y^{1}_{x}=y^{2}_{x}\omega\,\,\,\Rightarrow\,\,\,d\omega-y^{2}_{x}\omega=0$

As $\delta\omega=\delta y^{1}\wedge\delta y^{2}$, we obtain
$\omega\wedge\delta\omega=0$ and one can thus use the analytic Frobénius
theorem with integrating factor $y^{1}$ in order to get
$\omega=y^{1}\delta(y^{2}-log(y^{1}))$.
If $a\neq 0$, say $a=1$, we have $\lambda=y^{1}{\mu}^{1}+{\mu}^{2}$ and obtain
the only CC:

$y^{1}{\mu}^{1}_{x}+2\,y^{1}_{x}{\mu}^{1}+{\mu}^{2}_{x}+y^{2}_{x}{\mu}^{2}=0$

Multiplying by a test function $\xi$ and integrating by parts, we obtain the
parametrization:

$-y^{1}d{\xi}+y^{1}_{x}{\xi}={\eta}^{1},\hskip
56.9055pt-d{\xi}+y^{2}_{x}{\xi}={\eta}^{2}$

which is injective with potential $\xi=y^{1}{\eta}^{2}-{\eta}^{1}$.

EXAMPLE 3.3.9: With $n=1,m=3,K=\mathbb{Q}$, let us consider the first order
nonlinear system ([21]):

$P\equiv 2y^{3}_{x}+(y^{2}_{x})^{2}-(y^{1}_{x})^{2}=0$

The differenial ideal $\mathfrak{p}$ is prime because we have
$K\\{y\\}/\mathfrak{p}=K[y^{1},y^{1}_{x},...;y^{2},y^{2}_{x},...;y^{3}]$ is an
integral domain and we define as usual the differential extension
$L=Q(L\\{y\\}/\mathfrak{p})$.
Setting $\delta y=\eta$ and dividing by $2$, the linearized system becomes:

${\eta}^{3}_{x}+y^{2}_{x}{\eta}^{2}_{x}-y^{1}_{x}{\eta}^{1}_{x}=0$

Multiplying as usual by the Lagrange multiplier $\lambda$ and integrating by
parts, we get the adjoint operator with $d=d_{x}$:

$\left\\{\begin{array}[]{rcl}{\eta}^{1}&\rightarrow&y^{1}_{x}d\lambda+y^{1}_{xx}\lambda={\mu}^{1}\\\
{\eta}^{2}&\rightarrow&-y^{2}_{x}d\lambda-y^{2}_{xx}\lambda={\mu}^{2}\\\
{\eta}^{3}&\rightarrow&-d\lambda={\mu}^{3}\end{array}\right.$

which is injective with the two CC:

$\frac{{\mu}^{1}+y^{1}_{x}{\mu}^{3}}{y^{1}_{xx}}+\frac{{\mu}^{2}-y^{2}_{x}{\mu}^{3}}{y^{2}_{xx}}=0,\hskip
56.9055ptd(\frac{{\mu}^{1}+y^{1}_{x}{\mu}^{3}}{y^{1}_{xx}})+{\mu}^{3}=0$

It follows that $M={\Omega}_{L/K}$ is a torsion-free differential module over
$L[d]$ which is thus also free because it is known that any module over a
principal ideal ring which is torsion-free is also free ([39]). Its adjoint
operator provides therefore the first order parametrization:

$\left\\{\begin{array}[]{rcl}{\mu}^{1}&\rightarrow&-\frac{1}{y^{1}_{xx}}d{\xi}^{2}-\frac{1}{y^{1}_{xx}}{\xi}^{1}={\eta}^{1}\\\
{\mu}^{2}&\rightarrow&\frac{1}{y^{2}_{xx}}{\xi}^{1}={\eta}^{2}\\\
{\mu}^{3}&\rightarrow&-\frac{y^{1}_{x}}{y^{1}_{xx}}d{\xi}^{2}+(\frac{y^{1}_{x}}{y^{1}_{xx}}-\frac{y^{2}_{x}}{y^{2}_{xx}}){\xi}^{1}+{\xi}^{2}={\eta}^{3}\end{array}\right.$

This parametrization is injective because
${\xi}^{1}=y^{2}_{xx}{\eta}^{2},\,\,\,\,{\xi}^{2}={\eta}^{3}+y^{2}_{x}{\eta}^{2}-y^{1}_{x}{\eta}^{1}$.
Hence, we can replace any solution $({\eta}^{1},{\eta}^{2},{\eta}^{3})$ of the
linearized system by any couple $({\xi}^{1},{\xi}^{2})$, a result not evident
at first sight.
Finally, considering the two parametrization vertical $1$-forms:

${\omega}^{1}=\frac{1}{y^{2}_{xx}}{\xi}^{1}={\eta}^{2}=\delta y^{2},\hskip
28.45274pt{\omega}^{2}=\delta y^{3}+y^{2}_{x}\delta y^{2}-y^{1}_{x}\delta
y^{1}$

we have:

$\delta{\omega}^{1}=0\Rightarrow{\omega}^{1}\wedge{\omega}^{2}\wedge\delta{\omega}^{1}=0,\hskip
28.45274pt{\omega}^{1}\wedge{\omega}^{2}\wedge\delta{\omega}^{2}=\delta
y^{1}\wedge\delta y^{2}\wedge\delta y^{3}\wedge\delta y^{1}_{x}\neq 0$

and cannot therefore use the Frobenius theorem in order to integrate this
vertical exterior system.
According to what has been said, the linear and the nonlinear systems are both
controllable. In particular, if the nonlinear system should not be
controllable, it means that there should exists at least one autonomous
element in $L$ that should be constrained by at least one OD equation. The
linearization of such an element should produce a torsion element in $L$. The
striking feature of this example is that one can prove that $L$ is a purely
differentially transcendental extension of the ground field $K$. Indeed, we
may rewrite the system like:

$d(2y^{3}-(y^{1}-y^{2})d(y^{1}+y^{2}))+(y^{1}-y^{2})d^{2}(y^{1}+y^{2})=0$

Setting:

$z^{1}=2y^{3}-(y^{1}-y^{2})(y^{1}_{x}+y^{2}_{x}),\hskip
28.45274ptz^{2}=y^{1}+y^{2}\hskip 28.45274pt\Rightarrow\hskip
28.45274pt-\frac{z^{1}_{x}}{z^{2}_{xx}}=y^{1}-y^{2}$

we obtain the second order nonlinear parametrization:

$2\,y^{1}=-\frac{z^{1}_{x}}{z^{2}_{xx}}+z^{2},\hskip
42.67912pt2\,y^{2}=\frac{z^{1}_{x}}{z^{2}_{xx}}+z^{2},\hskip
42.67912pt2\,y^{3}=-\frac{z^{1}_{x}z^{2}_{x}}{z^{2}_{xx}}+z^{1}$

and thus $L=K<z^{1},z^{2}>$. Hence, introducing ${\zeta}^{1}=\delta
z^{1},{\zeta}^{2}=\delta z^{2}$, we can similarly replace any solution
$({\eta}^{1},{\eta}^{2},{\eta}^{3})$ of the linearized system by any couple
$({\zeta}^{1},{\zeta}^{2})$, a result even less evident at first sight . The
new parametrization is also injective but, contrary to the previous situation,
we have now ${\bar{\omega}}=\zeta=\delta z\Rightarrow\delta{\bar{\omega}}=0$.
As a (difficult) exercise of formal integrability, we let the reader prove
that the second order $2\times 2$ operator matrix $\xi\rightarrow\zeta$ is an
isomorphism (See [21], p 821 for more details on this substitution).

EXAMPLE 3.3.10: With $n=2,m=3,q=1,K=\mathbb{Q}$, let us consider the
differential ideal $\mathfrak{p}$ generated by the two differential
polynomials:

$\hskip 56.9055ptP_{1}\equiv y^{1}_{2}-y^{3}y^{1}_{1}=0,\hskip
56.9055ptP_{2}\equiv y^{2}_{2}-y^{3}y^{2}_{1}=0$

The corresponding system is easily sen to be involutive and we have:

$K\\{y^{1},y^{2},y^{3}\\}/\mathfrak{p}=K\\{y^{3}\\}[y^{1},y^{1}_{1},...;y^{2},y^{2}_{1},...]$

as an integral domain and $\mathfrak{p}$ is thus a prime differential ideal
allowing to define the differential field $L=Q(K\\{y\\}/\mathfrak{p})$ with
differential transcendence basis $K<y^{3}>\subset L$.
The linearized system ${\cal{D}}_{1}\eta=0$ is:

$d_{2}{\eta}^{1}-y^{3}d_{1}{\eta}^{1}-y^{1}_{1}{\eta}^{3}=0,\hskip
56.9055ptd_{2}{\eta}^{2}-y^{3}d_{1}{\eta}^{2}-y^{2}_{1}{\eta}^{3}=0$

Multiplying the first by ${\lambda}^{1}$, the second by ${\lambda}^{2}$ and
integrating by parts, we obtain the adjoint operator
$ad({\cal{D}}_{1})\lambda=\mu$:

$\left\\{\begin{array}[]{rcl}{\eta}^{1}&\rightarrow&-d_{2}{\lambda}^{1}+y^{3}d_{1}{\lambda}^{1}+y^{3}_{1}{\lambda}^{1}={\mu}^{1}\\\
{\eta}^{2}&\rightarrow&-d_{2}{\lambda}^{2}+y^{3}d_{1}{\lambda}^{2}+y^{3}_{1}{\lambda}^{2}={\mu}^{2}\\\
{\eta}^{3}&\rightarrow&-y^{1}_{1}{\lambda}^{1}-y^{2}_{1}{\lambda}^{2}={\mu}^{3}\end{array}\right.$

However, this operator is not involutive because it is not even formally
integrable. Nevertheless, adding the first order PD equation obtained by
prolonging the zero order equation with respect to $x^{1}$, we obtain:

$\left\\{\begin{array}[]{rcl}{\eta}^{1}&\rightarrow&-d_{2}{\lambda}^{1}+y^{3}d_{1}{\lambda}^{1}+y^{3}_{1}{\lambda}^{1}={\mu}^{1}\\\
{\eta}^{2}&\rightarrow&-d_{2}{\lambda}^{2}+y^{3}d_{1}{\lambda}^{2}+y^{3}_{1}{\lambda}^{2}={\mu}^{2}\\\
&&-y^{1}_{1}d_{1}{\lambda}^{1}-y^{2}_{1}d_{1}{\lambda}^{2}-y^{1}_{11}{\lambda}^{1}-y^{2}_{11}{\lambda}^{2}=d_{1}{\mu}^{3}\\\
{\eta}^{3}&\rightarrow&-y^{1}_{1}{\lambda}^{1}-y^{2}_{1}{\lambda}^{2}={\mu}^{3}\end{array}\right.\framebox{
$\begin{array}[]{cc}1&2\\\ 1&2\\\ 1&\bullet\\\ \bullet&\bullet\end{array}$ }$

We obtain the unique generating first order CC $ad({\cal{D}})\mu=0$, namely:

$d_{2}{\mu}^{3}-y^{3}d_{1}{\mu}^{3}-y^{1}_{1}{\mu}^{1}-y^{2}_{1}{\mu}^{2}-2y^{3}_{1}{\mu}^{3}=0$

Multiplying this CC by the test function $\xi$ and integrating by parts, we
get ${\cal{D}}\xi=\eta$ over $L$:

$\left\\{\begin{array}[]{rcl}{\mu}^{1}&\rightarrow&-y^{1}_{1}\xi={\eta}^{1}\\\
{\mu}^{2}&\rightarrow&-y^{2}_{1}\xi={\eta}^{2}\\\
{\mu}^{3}&\rightarrow&-d_{2}\xi+y^{3}d_{1}\xi-y^{3}_{1}\xi={\eta}^{3}\end{array}\right.$

Sustituting, we check the two CC described by ${\cal{D}}_{1}\eta=0$ plus an
additional zero order CC providing the torsion element $\omega=y^{1}_{1}\delta
y^{2}-y^{2}_{1}\delta y^{1}$ generating $ext^{1}(M)$ as we have indeed
$d_{2}\omega-y^{3}d_{1}\omega-y^{3}_{1}\omega=0$. We let the reader check that
$\omega\wedge\delta\omega\neq 0$. It follows that $\omega$ cannot have any
integrating factor according to the Frobenius Theorem.
Eliminating $y^{3}$, we are left with $(y^{1},y^{2})$ and the nonlinear system
$y^{1}_{1}y^{2}_{2}-y^{1}_{2}y^{2}_{1}=0$ with the same conlusions as before.
On the contrary, we let the reader check that the module of Kähler
differentials is torsion-free for the nonlinear system
$y^{1}_{1}y^{2}_{2}-y^{1}_{2}y^{2}_{1}=1$.

4) CONCLUSION

The author of this paper got his PhD thesis under the supervising of Prof. A.
Lichnerowicz and has been collaborating with him till his death in 1998. At
the end of his life, he became more and more convinced that the variational
origin of mathematical physics (elasticity, electromagnetism, general
relativity) through the corresponding Euler-Lagrange equations was a kind of
”screen ” hiding a more important concept allowing to describe the ”duality”
existing between ”fields ” and ”inductions ”. After the author discovered in
1995 the impossibility to parametrize the Einstein operator along a challenge
proposed by J. Wheeler in 1970, he did notice that, in control theory, ”a
control system is controllable if and only if it is parametrizable ” and that
the ”screen ” is just the ”differential double duality ” involved in
differential homological algebra through the use of the ”extension modules ”
(See Zbl 1079.93001). Hence it remained to study the systematic use of the
formal adjoint in the noncommutative situation met when linearizing nonlinear
systems of OD or PD equations. Among the best useful examples, one has the
following differential sequence, indicating below the fiber dimensions of the
vector bundles involved with $F_{0}=S_{2}T^{*}$:

$0\rightarrow\Theta\rightarrow T\stackrel{{\scriptstyle
Killing}}{{\longrightarrow}}F_{0}\stackrel{{\scriptstyle
Riemann}}{{\longrightarrow}}F_{1}\stackrel{{\scriptstyle
Bianchi}}{{\longrightarrow}}F_{2}\rightarrow...$
$0\rightarrow\Theta\rightarrow n\rightarrow n(n+1)/2\rightarrow
n^{2}(n^{2}-1)/12\rightarrow n^{2}(n^{2}-1)(n-2)/24\rightarrow...$

where $\Theta$ is the sheaf of Killing vector fields for the Euclidean metric
when $n=3$ in elasticity or the Minkowski metric when $n=4$ in general
relativity. Defining the adjoint operators:

$Cauchy=ad(Killing),\,\,Beltrami=ad(Riemann),\,\,Lanczos=ad(Bianchi)$

one discovers that Lanczos was in fact dreaming to construct the adjoint
differential sequence:

$0\leftarrow ad(T)\stackrel{{\scriptstyle
Cauchy}}{{\longleftarrow}}ad(F_{0})\stackrel{{\scriptstyle
Beltrami}}{{\longleftarrow}}ad(F_{1})\stackrel{{\scriptstyle
Lanczos}}{{\longleftarrow}}ad(F_{2})\leftarrow...$

where $ad(E)={\wedge}^{n}T^{*}\otimes E^{*}$ for any vector bundle $E$ where
$E^{*}$ is obtained from $E$ by inverting the transition rules when changing
local coordinates. Accordingly, the only true problem left was to prove that
each operator is indeed parametrized by the preceding one in both sequences, a
highly non evident fact. Similarly, we have the Poincaré sequence for the
exterior derivative:

$T^{*}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}{\wedge}^{2}T^{*}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}{\wedge}^{3}T^{*}$ $\hskip 42.67912ptn\rightarrow
n(n-1)/2\rightarrow n(n-1)(n-2)/6$

with $dA=F\Rightarrow dF=0$ for electromagnetism when $n=4$ where $A$ is the
EM potential and $F$ the EM field describing the first Maxwell operator and
its parametrization. The adjoint sequence:

$ad(T^{*})\stackrel{{\scriptstyle
ad(d)}}{{\longleftarrow}}ad({\wedge}^{2}T^{*})\stackrel{{\scriptstyle
ad(d)}}{{\longleftarrow}}ad({\wedge}^{3}T^{*})$

is used for the EM induction and second Maxwell operator both with its
parametrization by the so-called pseudo-potential. In both situations, there
is no need to appeal to variational calculus which is only used for exhibiting
the respective constitutive laws. We hope that the many tricky examples
presented in this paper will be used later on as test-examples for computer
algebra.

REFERENCES

[1] Aksteiner, S., Andersson, L., Bäckdahl, T.: New Identities for Linearized
Gravity on the Kerr Space-time, https://arxiv.org/1601.06084v3
[2] Bialynicki-Birula, A.: On Galois Theory of Fields with Operators, Amer.J.
Math., 84 (1962) 89-109.
[3] Bjork, J.E.: Analytic D-Modules and Applications, Kluwer (1993).
[4] Bourbaki, N.: Algèbre, Chapter 10, Algèbre Homologique, Springer, 2006.
[5] Cosserat, E., & Cosserat, F.: Théorie des Corps Déformables, Hermann,
Paris, (1909).
[6] Janet, M.: Sur les Systèmes aux Dérivées Partielles, Journal de Math., 8
(1920) 65-151.
[7] Johnson, J.: Kähler Differentials and Differential Algebra, Trans. Amer.
Math. Society, 192 (1974) 201- 208.
[8] Johnson, J.: Prolongations of Integral Domains, Journal of Algebra, 94
(1985) 173-210.
[9] Kaplansky, I.: An Introduction to Differential Algebra, Hermann, Paris
((1957, 1976).
[10] Kashiwara, M.: Algebraic Study of Systems of Partial Differential
Equations, Mémoires de la Société Mathématique de France, 63 (1995) (Transl.
from Japanese of his 1970 MasterÕs Thesis).
[11] Kolchin, E. R.: Differential Algebra and Algebraic Groups, Academic
Presss, (1973).
[12] Macaulay, F.S.: The Algebraic Theory of Modular Systems, Cambridge Tract
19, Cambridge University Press, London, 1916 (Reprinted by Stechert-Hafner
Service Agency, New York, 1964).
[13] Northcott, D.G.: An Introduction to Homological Algebra, Cambridge
university Press (1966).
[14] Oberst, U.: Multidimensional Constant Linear Systems, Acta Appl. Math.,
20 (1990) 1-175.
[15] Palamodov, V.P.: Linear Differential Operators with Constant
Coefficients, Grundlehren der Mathematischen Wissenschaften, 168, Springer-
Verlag (1970).
[16] Pirani, F.A.E., Robinson, D.C., Shadwick, W.F.: Local Jet Bundle
Formulation of Bäcklund Transformations, MPST 1, D. Deidel, Springer (1979).
[17] Pommaret, J.-F.: Systems of Partial Differential Equations and Lie
Pseudogroups, Gordon and Breach, New York (1978); Russian translation: MIR,
Moscow,(1983).
[18] Pommaret, J.-F.: Differential Galois Theory, Gordon and Breach, New York
(1983).
[19] Pommaret, J.-F.: Lie Pseudogroups and Mechanics, Gordon and Breach, New
York (1988).