# Normal Forms in Differential Galois Theory for the Classical Groups

Daniel Robertz  Lehrstuhl für Algebra und Zahlentheorie, RWTH Aachen
University, D–52056 Aachen, Germany<EMAIL_ADDRESS>and
Matthias Seiß  Institut für Mathematik, Universität Kassel, D–34109 Kassel,
Germany<EMAIL_ADDRESS>
###### Abstract.

Let $G$ be a classical group of dimension $d$ and let
$\boldsymbol{a}=(a_{1},\dots,a_{d})$ be differential indeterminates over a
differential field $F$ of characteristic zero with algebraically closed field
of constants $C$. Further let $A(\boldsymbol{a})$ be a generic element in the
Lie algebra $\mathfrak{g}(F\langle\boldsymbol{a}\rangle)$ of $G$ obtained from
parametrizing a basis of $\mathfrak{g}$ with the indeterminates
$\boldsymbol{a}$. It is known (cf. [Jua04]) that the differential Galois group
of $\boldsymbol{y}^{\prime}=A(\boldsymbol{a})\boldsymbol{y}$ over
$F\langle\boldsymbol{a}\rangle$ is $G(C)$. In this paper we construct a
differential field extension $\mathcal{L}$ of $F\langle\boldsymbol{a}\rangle$
such that the field of constants of $\mathcal{L}$ is $C$, the differential
Galois group of $\boldsymbol{y}^{\prime}=A(\boldsymbol{a})\boldsymbol{y}$ over
$\mathcal{L}$ is still the full group $G(C)$ and $A(\boldsymbol{a})$ is gauge
equivalent over $\mathcal{L}$ to a matrix in normal form which we introduced
in [Seib]. We also consider specializations of the coefficients of
$A(\boldsymbol{a})$.

##  1\. Introduction

In classical Galois theory there is the well-known construction of the general
polynomial equation over $\mathbb{Q}$ with Galois group the symmetric group
$S_{n}$. The coefficients of the general equation are (up to sign) the
elementary symmetric polynomials in $n$ indeterminates $x_{1}$, …, $x_{n}$
over $\mathbb{Q}$. Every algebraic extension of $\mathbb{Q}$ defined by a
polynomial $p(x)$ of degree $n$ is obtained as a specialization by
substituting the roots of $p(x)$ for $x_{1}$, …, $x_{n}$ in the general
equation. The case of the symmetric group was generalized by Emmy Noether in
[Noe17], which led to new problems in invariant theory (e.g., Noether
problem).

In an analogous way a general linear differential equation with differential
Galois group the general linear group $\mathrm{GL}_{n}(C)$ over an
algebraically closed field $C$ of characteristic zero is obtained as follows.
Extend the linear action of $\mathrm{GL}_{n}(C)$ on the vector whose
coordinates are differential indeterminates $y_{1}$, …, $y_{n}$ to the
differential field of rational functions $C\langle y_{1},\ldots,y_{n}\rangle$.
Introducing a new differential indeterminate $Y$ and denoting by
$w(y_{1},\ldots,y_{n})$ the Wronskian of $y_{1}$, …, $y_{n}$, the general
differential equation is given by (cf., e.g., [Mag94, Chapter 2])

(1)
$0\,=\,Y^{(n)}+c_{n-1}Y^{(n-1)}+\ldots+c_{0}Y\,:=\,\frac{w(Y,y_{1},\ldots,y_{n})}{w(y_{1},\ldots,y_{n})}.$

In fact, $c_{n-1},\ldots,c_{0}$ are differentially algebraically independent
generators of the fixed field of $C\langle y_{1},\ldots,y_{n}\rangle$ under
$\mathrm{GL}_{n}(C)$ (and recall that the elementary symmetric polynomials are
algebraically independent generators of the fixed field of
$\mathbb{Q}(x_{1},\ldots,x_{n})$ under $S_{n}$). Every Picard-Vessiot
extension of a differential field $F$, with field of constants $C$, defined by
a linear differential polynomial of order $n$ is obtained as a specialization
of (1) by substituting the linearly independent solutions $\eta_{1}$, …,
$\eta_{n}$ for $y_{1}$, …, $y_{n}$. Generalizations to groups other than
$\mathrm{GL}_{n}(C)$ were obtained in [Gol57] and [JM07]. In all these cases
the general differential equation involves $n$ differential indeterminates
over $C$ (apart from $Y$).

A different approach resulting in general differential equations involving
$d=\dim(G)$ differential indeterminates was introduced by Lourdes Juan in
[Jua04] for connected linear algebraic groups $G$. Fixing a representation of
$G\subset\mathrm{GL}_{n}$ the general differential equation in this case is
the matrix differential equation defined by a general element
$A(\boldsymbol{a})$ of the Lie algebra $\mathfrak{g}$ of $G$ with differential
indeterminates $\boldsymbol{a}=(a_{1},\ldots,a_{d})$ over $C$ as coefficients.
It is shown that the function field
$\mathcal{E}_{J}=C\langle\boldsymbol{a}\rangle(G)$ of the group
$G(C\langle\boldsymbol{a}\rangle)$ of $C\langle\boldsymbol{a}\rangle$-rational
points is a Picard-Vessiot extension of $C\langle\boldsymbol{a}\rangle$ for
this equation with differential Galois group $G(C)$. The $d$ algebraically
independent rational functions generating $\mathcal{E}_{J}$ over
$C\langle\boldsymbol{a}\rangle$ as a field are differentially algebraically
independent over $C$ and generate $\mathcal{E}_{J}$ as a differential field
over $C$. Both fields $\mathcal{E}_{J}$ and $C\langle\boldsymbol{a}\rangle$
are purely differentially transcendental extensions of $C$ of degree $d$. This
implies that her construction is generic in the following sense. For a
differential field $F$ with field of constants $C$ suppose that
$A\in\mathfrak{gl}_{n}(F)$ defines a Picard-Vessiot extension of $F$ with
differential Galois group a connected subgroup of $G(C)$. Then there are
elements $\boldsymbol{f}=(f_{1},\dots,f_{d})$ in $F$ such that the matrix $A$
is gauge equivalent to the matrix $A(\boldsymbol{f})$ obtained by specializing
the coefficients $\boldsymbol{a}$ in $A(\boldsymbol{a})$ to $\boldsymbol{f}$.

A third approach to constructing general differential equations for the
classical groups was presented in [Seib]. This approach combines the geometric
structure of a classical group $G$ of Lie rank $l$ with Picard-Vessiot theory
and involves only $l$ differential indeterminates. More precisely, the
construction starts with a differential field $C\langle\mathbf{v}\rangle$
generated by $l$ differential indeterminates $\mathbf{v}=(v_{1},\dots,v_{l})$
over $C$. The general extension field in this approach is a Liouvillian
extension $\mathcal{E}$ of $C\langle\mathbf{v}\rangle$ with differential
Galois group a fixed Borel subgroup $B^{-}(C)$ of $G(C)$. As in the case of
$\mathrm{GL}_{n}(C)$ we construct a fundamental matrix $Y$ and define an
action of $G(C)$ on it which will then induce an action on $\mathcal{E}$.
Fixing a Chevalley basis of $\mathfrak{g}$, the defining matrix of
$\mathcal{E}$ is chosen such that its conjugate by a representative
$\overline{w}$ of the longest Weyl group element is the sum of the Cartan
subalgebra, parametrized by $\mathbf{v}$, and the basis elements of the root
spaces corresponding to the simple roots. Choosing a fundamental matrix $b\in
B^{-}(\mathcal{E})$ generating $\mathcal{E}$ over $C\langle\mathbf{v}\rangle$,
we can construct a matrix $u$ in the maximal unipotent subgroup of
$B^{-}(C\langle\mathbf{v}\rangle)$ such that the logarithmic derivative of
$Y=u\overline{w}b$ is the matrix $A_{G}(\mathbf{s})$ constructed in [Seib] and
[Seia]. The differential polynomials $\mathbf{s}=(s_{1},\dots,s_{l})$ in
$C\\{\mathbf{v}\\}$ are differentially algebraically independent over $C$. The
matrix $u$ is the product of matrices of root groups corresponding to all
negative roots $\Phi^{-}$ and it depends on $|\Phi^{-}|$ differential
polynomials $\boldsymbol{p}$ in $C\\{\mathbf{v}\\}$, those corresponding to
the negative simple roots being the indeterminates $\mathbf{v}$. Multiplying
$Y$ from the right by an element of the full group $G(C)$ and then taking the
Bruhat decomposition defines an action on $\boldsymbol{p}$ and on the
generators of the Liouvillian extension, i.e. the entries of $b$, and
therefore on $\mathcal{E}$. The fixed field under the induced action of $G(C)$
on $\mathcal{E}$ is $C\langle\mathbf{s}\rangle$ and it is shown that the
extension $\mathcal{E}$ of $C\langle\mathbf{s}\rangle$ is a Picard-Vessiot
extension with differential Galois group $G(C)$. The construction is only
generic for Picard-Vessiot extensions of $F$ with defining matrix gauge
equivalent to a matrix in _normal form_ , i.e. a specialization of
$A_{G}(\mathbf{s})$. Deciding such a gauge equivalence is non-trivial as a
consequence of the fact that $\mathcal{E}$ and $C\langle\mathbf{s}\rangle$
have differential transcendence degree $l$ over $C$.

The purpose of this paper is to contribute to the question of the genericity
properties of the general extension $\mathcal{E}$ over
$C\langle\mathbf{s}\rangle$. We study the relationship of the two general
extensions $\mathcal{E}/C\langle\mathbf{s}\rangle$ and
$\mathcal{E}_{J}/C\langle\boldsymbol{a}\rangle$ in detail. Clearly the matrix
$A_{G}(\mathbf{s})$ can be obtained by a specialization of $A(\boldsymbol{a})$
and so Juan’s general extension
$\mathcal{E}_{J}/C\langle\boldsymbol{a}\rangle$ is generic for
$\mathcal{E}/C\langle\mathbf{s}\rangle$. For the opposite direction we need to
solve the problem of gauge equivalence of $A(\boldsymbol{a})$ to a matrix in
normal form. Over the differential field $C\langle\boldsymbol{a}\rangle$ this
gauge equivalence does not hold in general111A simple counterexample is the
case $G=\mathrm{SL}_{2}$, where gauge transformation of $A(\boldsymbol{a})$
into normal form is only possible over an extension $\mathcal{L}$ of
$C\langle\boldsymbol{a}\rangle$, for example, a quadratic extension..
Therefore we construct a differential extension field allowing gauge
transformation into normal form and preserving the differential Galois group.
Our main theorem is

######  Theorem 1.1 (Theorem 6.7).

There is a differential extension field $\mathcal{L}$ of
$C\langle\boldsymbol{a}\rangle$ with field of constants $C$ such that
$A(\boldsymbol{a})$ is gauge equivalent over $\mathcal{L}$ to a specialization
of $A_{G}(\mathbf{s})$ and the differential Galois group of
$A(\boldsymbol{a})$ over $\mathcal{L}$ is the full group $G(C)$.

$\mathcal{E}_{J}$$C\langle\boldsymbol{a}\rangle$$\mathcal{E}_{J}\cdot\mathcal{L}$$\mathcal{L}$$\mathcal{E}$$C\langle\mathbf{s}\rangle$$G(C)$$G(C)$$G(C)$
Figure 1. Relation between extensions $\mathcal{E}/C\langle\mathbf{s}\rangle$
and $\mathcal{E}_{J}\cdot\mathcal{L}/\mathcal{L}$ via specialization

According to the theorem we obtain, after translating the generic extension
$\mathcal{E}_{J}/C\langle\boldsymbol{a}\rangle$ by $\mathcal{L}$, the Picard-
Vessiot extension $\mathcal{E}_{J}\cdot\mathcal{L}/\mathcal{L}$ as a
specialization of the general extension
$\mathcal{E}/C\langle\mathbf{s}\rangle$. But this does not imply that the
genericity properties of $\mathcal{E}_{J}/C\langle\boldsymbol{a}\rangle$ carry
over to $\mathcal{E}/C\langle\mathbf{s}\rangle$. A reason for this is that, in
contrast to $C\langle\boldsymbol{a}\rangle$, the extension $\mathcal{L}$ is
not purely differentially transcendental over $C$ and so specializations of
the extension $\mathcal{E}_{J}\cdot\mathcal{L}/\mathcal{L}$ need to respect
the differential algebraic relations of the generators of $\mathcal{L}$. We
address how one may overcome this problem in case of a specialization $E/F$ of
$\mathcal{E}_{J}/C\langle\boldsymbol{a}\rangle$ with differential Galois group
$G(C)$. We present in Theorem 8.1 a sufficient condition on a defining matrix
in $\mathfrak{g}(F)$ ensuring that there exists a differential extension field
$L$ of $F$ with field of constants $C$ such that $E\cdot L/L$ is a
specialization of $\mathcal{E}/C\langle\mathbf{s}\rangle$ with the same
differential Galois group. In case of $\mathrm{SL}_{3}$ we show how one can
guarantee that this condition is satisfied.

More recently, Man Cheung Tsui has shown in [Tsu] that the differential
transcendence degree of a generic Picard-Vessiot extension for $G$ with
constants $C$ is bounded below by the essential dimension of any Picard-
Vessiot extension with differential Galois group $G$ and the same field of
constants. More precisely, let $E/F$ and $E_{0}/F_{0}$ be Picard-Vessiot
extensions with differential Galois group $G(C)$ such that $E_{0}\subset E$
and $F_{0}\subset F$. If $F\cdot E_{0}=E$, then $F_{0}$ is called a
differential field of definition for $E/F$. If $A$ is a defining matrix for
$E/F$, then the differential field generated over $C$ by the entries of any
gauge equivalent matrix is a differential field of definition of $E/F$. The
essential dimension is now defined as the smallest differential transcendence
degree over $C$ of a field of definition $F_{0}\subseteq F$ of the extension.
Enlarging $F$ without changing the differential Galois group might have the
effect that the essential dimension drops. Our concept of genericity features
this possibility, leaving it open whether
$\mathcal{E}/C\langle\mathbf{s}\rangle$ is generic in our sense, while Man
Cheung Tsui’s results show that, as expected,
$\mathcal{E}/C\langle\mathbf{s}\rangle$ is not generic in the sense of [Tsu,
Def. 5.1].

After recalling the structure of the classical groups and fixing notation in
Section 2, basic properties of the gauge transformation are summarized in
Section 3. We investigate the possibility of gauge transformation to normal
form by using differential elimination in Section 5. For that reason we recall
the fundamental notion of Thomas decomposition in Section 4. The construction
of the field $\mathcal{L}$ which allows gauge transformation to normal form is
carried out in Section 6 and the existence of an intermediate differential
field $C\langle\boldsymbol{a}\rangle\subset\mathcal{L}_{2}\subset\mathcal{L}$
is proved such that $\mathcal{L}$ is generated over $\mathcal{L}_{2}$ by the
coefficients $\mathbf{\overline{t}}$ of the normal form, which are
differentially algebraically independent over $\mathcal{L}_{2}$. In Section 7
we prove that, even after extending the coefficient field from
$C\langle\mathbf{v}\rangle$ to $F\langle\mathbf{v}\rangle$ in the construction
of [Seib], the differential field extension defined by the normal form has the
expected Galois group, which completes the proof of Theorem 1.1. In the final
section we generalize the earlier discussion of purely differentially
transcendental extensions to more general extensions by introducing a
condition ensuring that the analogously constructed differential field
extension has the correct subfield of constants and the correct Galois group.
We also demonstrate how this condition is met in the case $G=\mathrm{SL}_{3}$.

##  2\. The Structure of the Classical Groups

Let $G\subset\mathrm{GL}_{n}(C)$ be one of the classical groups of Lie rank
$l$. We denote by $\Phi$ the root system of the corresponding type of $G$ and
by $\Delta=\\{\alpha_{1},\dots,\alpha_{l}\\}$ a basis of $\Phi$ with simple
roots $\alpha_{i}$. Each root in $\Phi$ can be written as a
$\mathbb{Z}$-linear combination of the simple roots with all coefficients
negative or positive. This yields a disjoint decomposition of $\Phi$ into a
set of positive and negative roots, i.e.

$\Phi\,=\,\Phi^{+}\cup\Phi^{-}.$

For a root $\alpha\in\Phi$ we write $\operatorname{ht}(\alpha)$ for the height
of $\alpha$, i.e. the sum of all coefficients in the $\mathbb{Z}$-linear
combination with respect of $\Delta$. Further denote by $m$ the cardinalities
of $\Phi^{+}$ and $\Phi^{-}$. We enumerate the positive roots as $\alpha_{1}$,
…, $\alpha_{l}$, $\alpha_{l+1}$, …, $\alpha_{m}$ in such a way that
$\alpha_{1}$, …, $\alpha_{l}$ are the simple roots introduced above and
$\operatorname{ht}(\alpha_{r})\leq\operatorname{ht}(\alpha_{s})$ for all
$r\leq s$.

We denote by $\mathcal{W}$ the Weyl group of $\Phi$. It is generated by the
$l$ simple reflections $w_{\alpha_{i}}$ for the simple roots $\alpha_{i}$ of
$\Delta$. Every $w\in\mathcal{W}$ can be written as a product of simple
reflections and if this expression is minimal we call the number of simple
reflections the length $l(w)$ of $w$. The length coincides with the number of
positive roots $\alpha\in\Phi^{+}$ with $w(\alpha)$ negative. There is a
unique element $\overline{w}$ of $\mathcal{W}$ of maximal length and this
element maps all positive roots to negative ones and vice versa. We write
$\mathfrak{g}\subset\mathfrak{gl}_{n}(C)$ for the Lie algebra of $G$ and we
let $\mathfrak{h}$ be the Cartan subalgebra of $\mathfrak{g}$ consisting of
diagonal matrices. Let

$\mathfrak{g}\,=\,\mathfrak{h}\oplus\bigoplus_{\alpha\in\Phi}\mathfrak{g}_{\alpha}$

be a Cartan decomposition of $\mathfrak{g}$, where $\mathfrak{g}_{\alpha}$
denotes the one-dimensional root space corresponding to the root $\alpha$ of
$\Phi$. In accordance with this decomposition we fix a Chevalley basis

(2) $\\{H_{i},X_{\alpha}\mid 1\leq i\leq l,\ \alpha\in\Phi\\},$

where $\mathfrak{h}=\langle H_{1},\dots,H_{l}\rangle$ and
$\mathfrak{g}_{\alpha}=\langle X_{\alpha}\rangle$.

We denote by $T$ the maximal torus of $G$ with Lie algebra $\mathfrak{h}$ and
for a root $\alpha\in\Phi$ we denote by $U_{\alpha}$ the root group whose Lie
algebra is the root space $\mathfrak{g}_{\alpha}$. Let

$\exp\colon\mathfrak{g}_{\alpha}\rightarrow U_{\alpha},\
X_{\alpha}\mapsto\sum_{j\geq 0}\frac{1}{j!}X_{\alpha}^{j}$

be the exponential map from the root space $\mathfrak{g}_{\alpha}$ to the root
group $U_{\alpha}$ of $G$. For $x\in C$ we write $u_{\alpha}(x)$ for
$\exp(xX_{\alpha})$. Let $U^{+}$ (respectively $U^{-}$) be the maximal
unipotent subgroup whose Lie algebra $\mathfrak{u}^{+}$ (respectively
$\mathfrak{u}^{-}$) is the direct sum of all root spaces that are indexed by
the positive (respectively negative) roots. Further we denote by $B^{+}$
(respectively $B^{-}$) the Borel subgroup of $G$ with Lie algebra
$\mathfrak{b}^{+}=\mathfrak{h}\oplus\mathfrak{u}^{+}$ (respectively
$\mathfrak{b}^{-}=\mathfrak{h}\oplus\mathfrak{u}^{-}$). Clearly we have
$T<B^{+}$ and $B^{+}=TU^{+}$ (respectively $T<B^{-}$ and $B^{-}=TU^{-}$).
Moreover, to shorten notation we will omit the plus sign in the notation of
the subgroups and algebras, i.e., we will write $B$ for $B^{+}$, etc.

The following theorems provide a normal form for elements of $G$ parametrized
by $B$ and $\mathcal{W}$ (cf. [Hum75, 28.3 Theorem and 28.4 Theorem]).

######  Theorem 2.1 (Bruhat decomposition).

We have $G=\bigcup_{w\in\mathcal{W}}BwB$ (disjoint union) with
$BwB=B\widetilde{w}B$ if and only if $w=\widetilde{w}$ in $\mathcal{W}$.

For each $w\in\mathcal{W}$ fix a coset representative $n(w)$ in the normalizer
of $T$ in $G$.

######  Theorem 2.2.

Each element $g\in G$ can be written in the form

$g\,=\,u^{\prime}\,n(w)\,t\,u\,,$

where $w\in\mathcal{W}$, $t\in T$, $u\in U$ and $u^{\prime}\in
U^{\prime}_{w}=U\cap n(w)U^{-}n(w)^{-1}$ are all determined uniquely by $g$.

In the following we will consider the group $G$ over different fields. For a
field $K$ of characteristic zero, $G(K)$ is the group of $K$-rational points.

##  3\. Gauge Equivalence

For $g\in G\subset\mathrm{GL}_{n}$ we denote by $\operatorname{Ad}(g)$ the
automorphism

$\operatorname{Ad}(g):\mathfrak{g}\rightarrow\mathfrak{g},\ X\mapsto
gXg^{-1},$

i.e., $\operatorname{Ad}$ is the adjoint action of $G$ on its Lie algebra
$\mathfrak{g}$. Moreover, for a differential field $F$ with derivation
$\partial_{F}$ the logarithmic derivative $\ell\delta$ is defined by

$\ell\delta\colon\mathrm{GL}_{n}(F)\rightarrow\mathfrak{gl}_{n}(F),\
g\mapsto\partial_{F}(g)g^{-1}.$

Two matrices $A_{1}$ and $A_{2}$ in $\mathfrak{g}(F)$ are _gauge equivalent_
over $F$ by $g\in G(F)$ if

$g.A_{1}:=\operatorname{Ad}(g)(A_{1})+\ell\delta(g)=A_{2}.$

The action of $G(F)$ on $\mathfrak{g}(F)$ by gauge transformation partitions
$\mathfrak{g}(F)$ into orbits depending on the differential field $F$. We will
use the following two remarks to describe the images of the adjoint action and
the logarithmic derivative, respectively. Together they enable us to connect
gauge transformation of an element of $\mathfrak{g}$ by a root group element
with the root structure.

######  Remark 3.1.

For linearly independent $\alpha$, $\beta\in\Phi$ and $r,q\in\mathbb{N}$ let
$\alpha-r\beta,\dots,\alpha+q\beta$ be the $\beta$-string through $\alpha$ and
let $\langle\alpha,\beta\rangle$ be the Cartan integer. Then, with respect to
the Chevalley basis defined in (2), we have [Car89, Section 4.3]

$\displaystyle\mathrm{Ad}(u_{\beta}(x))(X_{\alpha})$ $\displaystyle=$
$\displaystyle\sum\nolimits_{i=0}^{q}c_{\beta,\alpha,i}x^{i}X_{\alpha+i\beta},$
$\displaystyle\mathrm{Ad}(u_{\beta}(x))(H_{\alpha})$ $\displaystyle=$
$\displaystyle H_{\alpha}-\langle\alpha,\beta\rangle X_{\beta},$
$\displaystyle\mathrm{Ad}(u_{\beta}(x))(X_{-\beta})$ $\displaystyle=$
$\displaystyle X_{-\beta}+xH_{\beta}-x^{2}X_{\beta},$

where $c_{\beta,\alpha,0}=1$ and $c_{\beta,\alpha,i}=\pm\binom{r+i}{i}$.

######  Remark 3.2.

Let $G\subset\mathrm{GL}_{n}$ be a linear algebraic group. Then the
restriction of $\ell\delta$ to $G$ maps $G(F)$ to its Lie algebra
$\mathfrak{g}(F)$, i.e., we have

$\ell\delta|_{G}:G(F)\rightarrow\mathfrak{g}(F).$

A proof can be found in [Kov69].

##  4\. Thomas Decomposition of Differential Systems

In this section we recall the notion of Thomas decomposition of differential
systems to the extent it will be used in later sections. For more details we
refer to, e.g., [Rob14].

The Bruhat decomposition of classical groups plays a crucial role in our work.
We shall split the task of deciding gauge equivalence to a matrix in normal
form into checking consistency of differential systems associated to the
Bruhat cells of $G$ (cf. Section 5 for the definition of these differential
systems $S_{w}$ and Definition 6.1 for the definition of normal form). A
different application of Thomas decomposition to the Bruhat decomposition of
the general linear group was presented in [Ple09], where the Bruhat cells are
shown to be described by simple algebraic systems.

Let $F$ be a differential field of characteristic zero and
$R=F\\{u_{1},\ldots,u_{n}\\}$ the differential polynomial ring over $F$ in the
differential indeterminates $u_{1}$, …, $u_{n}$. Differential polynomials in
$u_{1}$, …, $u_{n}$ are polynomials in the algebraically independent
derivatives

$\\{\partial^{j}u_{i}\mid i\in\\{1,\ldots,n\\},\,j\in\mathbb{Z}_{\geq 0}\\},$

with coefficients in $F$, where the derivation $\partial$ on $R$ extends the
derivation on $F$. We endow $R$ with a _ranking_ $>$, i.e. a total ordering
$>$ on the above set such that $\partial^{j}u_{i}>u_{i}$ for all $i$ and all
$j\geq 1$, and for all $j_{1}$, $j_{2}\in\mathbb{Z}_{\geq 0}$, the relation
$\partial^{j_{1}}u_{i_{1}}>\partial^{j_{2}}u_{i_{2}}$ implies
$\partial^{j_{1}+k}u_{i_{1}}>\partial^{j_{2}+k}u_{i_{2}}$ for all $k\geq 0$. A
ranking $>$ is said to be _orderly_ if $j_{1}>j_{2}$ implies
$\partial^{j_{1}}u_{i_{1}}>\partial^{j_{2}}u_{i_{2}}$ for all $i_{1}$,
$i_{2}\in\\{1,\ldots,n\\}$.

######  Definition 4.1.

A _differential system_ $S$, defined over $R=F\\{u_{1},\ldots,u_{n}\\}$, is
given by finitely many equations and inequations

(3) $p_{1}=0,\quad p_{2}=0,\quad\ldots,\quad p_{s}=0,\quad q_{1}\neq 0,\quad
q_{2}\neq 0,\quad\ldots,\quad q_{t}\neq 0,$

where $p_{1}$, …, $p_{s}$, $q_{1}$, …, $q_{t}\in R$ and $s$,
$t\in\mathbb{Z}_{\geq 0}$. For a given open subset $\Omega$ of $\mathbb{C}$
the _solution set_ of $S$ is

$\displaystyle\operatorname{Sol}_{\Omega}(S):=\\{\,f=(f_{1},\ldots,f_{n})$
$\displaystyle|$ $\displaystyle f_{k}\colon\Omega\to\mathbb{C}\mbox{
analytic},\,k=1,\ldots,n,$ $\displaystyle p_{i}(f)=0,\,q_{j}(f)\neq
0,\,i=1,\ldots,s,\,j=1,\ldots,t\,\\}.$

For each differential polynomial $p\in R\setminus F$ the _leader_ of $p$ is
the derivative $\partial^{j}u_{i}$ that is ranked highest with respect to $>$
among those occurring in $p$. Differential polynomials are considered
recursively as polynomials in their leaders with coefficients that are
differential polynomials in lower ranked derivatives, etc. The _initial_ of
$p$ is the leading coefficient of $p$ as a polynomial in its leader. The
_separant_ of $p$ is the partial derivative of $p$ with respect to its leader.

By using the above concepts, besides the structure of $R$ as differential
ring, pseudo-reduction of one differential polynomial by another can be
defined, if the leaders of both differential polynomials involve the same
differential indeterminate, which results in a pseudo-remainder (cf., e.g.,
[Rob14] for details).

######  Definition 4.2.

A differential system $S$, defined over $R$, as in (3) is said to be _simple_
(with respect to $>$) if the following three conditions hold.

  1. (a)

The system $S$ is simple as an algebraic system. More precisely, we consider
the finitely many derivatives $\partial^{j}u_{i}$ which occur in the equations
and inequations of $S$ as coordinates of an affine space $\mathbb{A}$ over an
algebraic closure of $F$, totally ordered by $>$. Then we require that:

    1. (i)

$p_{i}\not\in F$ for all $i$ and $q_{j}\not\in F$ for all $j$;

    2. (ii)

the leaders of $p_{1}$, …, $p_{s}$, $q_{1}$, …, $q_{t}$ are pairwise distinct;

    3. (iii)

no initial or separant of $p_{1}$, …, $p_{s}$, $q_{1}$, …, $q_{t}$ has a zero
in the algebraic set defined by $p_{1}=0$, …, $p_{s}=0$, $q_{1}\neq 0$, …,
$q_{t}\neq 0$ in $\mathbb{A}$.

  2. (b)

No pseudo-reduction of any $p_{i}$ modulo $p_{1}$, …, $p_{i-1}$, $p_{i+1}$, …,
$p_{s}$ and their derivatives is possible.

  3. (c)

The left hand sides of the inequations $q_{1}\neq 0$, …, $q_{t}\neq 0$ equal
their pseudo-remainders modulo $p_{1}$, …, $p_{s}$ and their derivatives.

######  Definition 4.3.

Let $S$ be a differential system, defined over $R$. A _Thomas decomposition_
of $S$ (or of $\operatorname{Sol}_{\Omega}(S)$) with respect to $>$ is a
collection of finitely many simple differential systems $S_{1}$, …, $S_{r}$,
defined over $R$, such that $\operatorname{Sol}_{\Omega}(S)$ is the disjoint
union of the solution sets $\operatorname{Sol}_{\Omega}(S_{1})$, …,
$\operatorname{Sol}_{\Omega}(S_{r})$.

If the differential system $S$ is defined over a computable differential ring,
then the construction of a Thomas decomposition of $S$ is algorithmic. Then
membership to the radical differential ideal generated by a simple
differential system can be decided effectively.

######  Proposition 4.4 ([Rob14], Prop. 2.2.50).

Let $S$ be a simple differential system, defined over $R$, with equations
$p_{1}=0$, …, $p_{s}=0$. Moreover, let $E$ be the differential ideal of $R$
generated by $p_{1}$, …, $p_{s}$ and define the product $q$ of the initials
and separants of all $p_{1}$, …, $p_{s}$. Then $E:q^{\infty}$ is a radical
differential ideal. Given $p\in R$, we have $p\in E:q^{\infty}$ if and only if
the pseudo-remainder of $p$ modulo $p_{1}$, …, $p_{s}$ and their derivatives
is zero.

More generally, if the differential system $S$ is not necessarily simple, we
have the following.

######  Proposition 4.5 ([Rob14], Prop. 2.2.72).

Let $S$ be a (not necessarily simple) differential system as in (3) and
$S_{1}$, …, $S_{r}$ a Thomas decomposition of $S$ with respect to any ranking
on $R$. Moreover, let $E$ be the differential ideal of $R$ which is generated
by $p_{1}$, …, $p_{s}$ and define the product $q$ of $q_{1}$, …, $q_{t}$. For
$i\in\\{1,\ldots,r\\}$, let $E^{(i)}$ be the differential ideal of $R$
generated by the equations in $S_{i}$ and define the product $q^{(i)}$ of the
initials and separants of all these equations. Then we have

$\sqrt{E:q^{\infty}}\,=\,\left(E^{(1)}:(q^{(1)})^{\infty}\right)\cap\ldots\cap\left(E^{(r)}:(q^{(r)})^{\infty}\right).$

In what follows, differential systems, defined over $R$, that are simple with
respect to a block ranking $>$ on $R$ will be considered. More precisely, let
$\\{u_{1},\ldots,u_{n}\\}=B_{1}\cup\ldots\cup B_{k}$ be a partition of the set
of differential indeterminates. For any $i\geq 1$ let
$F\\{B_{i},\ldots,B_{k}\\}$ be the differential polynomial ring over $F$ in
the differential indeterminates $B_{i}\cup\ldots\cup B_{k}$, considered as a
differential subring of $R=F\\{u_{1},\ldots,u_{n}\\}$. We refer to $>$ as an
_elimination ranking with blocks $B_{1}$, …, $B_{k}$_ if any derivative
involving a differential indeterminate in $B_{j}$ is ranked higher with
respect to $>$ than any derivative involving a differential indeterminate in
$B_{j+1}\cup\ldots\cup B_{k}$. Then we also write $B_{1}\gg B_{2}\gg\ldots\gg
B_{k}$.

######  Proposition 4.6 ([Rob14], Prop. 3.1.36).

Let $S$ be a differential system as in (3), defined over $R$, that is simple
with respect to an elimination ranking $>$ as defined above. Moreover, let $E$
be the differential ideal of $R$ generated by $p_{1}$, …, $p_{s}$ and $q$ the
product of the initials and separants of all $p_{1}$, …, $p_{s}$. For every
$i\in\\{1,\ldots,k\\}$, let $E_{i}$ be the differential ideal of
$F\\{B_{i},\ldots,B_{k}\\}$ generated by

$P_{i}:=\\{p_{1},\ldots,p_{s}\\}\cap F\\{B_{i},\ldots,B_{k}\\}$

and let $q_{i}$ be the product of the initials and separants of all elements
of $P_{i}$. Then, for every $i\in\\{1,\ldots,k\\}$, we have

$(E:q^{\infty})\cap F\\{B_{i},\ldots,B_{k}\\}=E_{i}:q_{i}^{\infty}\,.$

##  5\. Resolving Weyl Group Elements and their Associated Differential
Systems

Let $\mathbf{a}^{+}=(a^{+}_{1},\dots,a^{+}_{m})$,
$\mathbf{a}^{0}=(a^{0}_{1},\dots,a^{0}_{l})$ and
$\mathbf{a}^{-}=(a^{-}_{1},\dots,a^{-}_{m})$ be differential indeterminates
over $C$ and, using the notation introduced in Section 2, define a general
element of the Lie algebra $\mathfrak{g}$ by

(4)
$A:=\sum_{i=1}^{m}a^{+}_{i}X_{\alpha_{i}}+\sum_{i=1}^{m}a^{-}_{i}X_{-\alpha_{i}}+\sum_{i=1}^{l}a^{0}_{i}H_{i}.$

For further differential indeterminates $\mathbf{b}=(b_{1},\dots,b_{m})$ over
$C$ let $u_{-\alpha_{i}}(b_{i})$ be the image of $b_{i}X_{-\alpha_{i}}$ under
the exponential map, that is $u_{-\alpha_{i}}(b_{i})$ is a parametrization of
the root group $U_{-\alpha_{i}}$. We fix $w\in\mathcal{W}$. Let
$i_{1},\dots,i_{k}$ be the indices such that

$w(\Phi^{+})\cap\Phi^{-}=\\{-\alpha_{i_{1}},\dots,-\alpha_{i_{k}}\\},$

i.e., exchanging the roles played by $U^{+}$ and $U^{-}$ in Theorem 2.2,
$(U^{-})_{w}^{\prime}$ is generated by the root groups
$U_{-\alpha_{i_{1}}},\dots,U_{-\alpha_{i_{k}}}$. We set
$\mathbf{b}_{w}=(b_{i_{1}},\dots,b_{i_{k}})$ and define

(5) $u_{w}(\mathbf{b}_{w})=u_{-\alpha_{i_{1}}}(b_{i_{1}})\cdots
u_{-\alpha_{i_{k}}}(b_{i_{k}}).$

Gauge transformation of $A$ by $n(w)u_{w}(\mathbf{b}_{w})$ yields

$\begin{array}[]{rcl}(n(w)u_{w}(\mathbf{b}_{w})).A&=&\operatorname{Ad}(n(w)u_{w}(\mathbf{b}_{w}))(A)+\ell\delta(n(w)u_{w}(\mathbf{b}_{w}))\\\\[5.0pt]
&=&\displaystyle\sum_{i=1}^{m}f^{+}_{w,i}X_{\alpha_{i}}+\sum_{i=1}^{m}f^{-}_{w,i}X_{-\alpha_{i}}+\sum_{i=1}^{l}f^{0}_{w,i}H_{i}\,,\end{array}$

where $f^{+}_{w,i}$, $f^{-}_{w,i}$ and $f^{0}_{w,i}$ are certain differential
polynomials in
$C\\{\mathbf{a}^{+},\mathbf{a}^{-},\mathbf{a}^{0},\mathbf{b}_{w}\\}$. We
define the differential system $S_{w}$ by

$S_{w}:\quad f^{+}_{w,l+1}=0,\quad\dots,\quad f^{+}_{w,m}=0.$

The system expresses conditions for the vanishing of the coefficients of the
basis elements $X_{\alpha_{l+1}},\dots,X_{\alpha_{m}}$, i.e., modulo the
differential ideal $I_{w}$ of
$C\\{\mathbf{a}^{+},\mathbf{a}^{-},\mathbf{a}^{0},\mathbf{b}_{w}\\}$ generated
by $f^{+}_{w,l+1}$, …, $f^{+}_{w,m}$ the matrix $A$ will be gauge equivalent
to

$\sum_{i=1}^{l}\overline{f}^{+}_{w,i}X_{\alpha_{i}}+\sum_{i=1}^{m}\overline{f}^{-}_{w,i}X_{-\alpha_{i}}+\sum_{i=1}^{l}\overline{f}^{0}_{w,i}H_{i},$

where $\overline{f}^{\pm}_{w,i}$, $\overline{f}^{0}_{w,i}$ denote the images
of $f^{\pm}_{w,i}$, $f^{0}_{w,i}$ in the residue class ring.

The differential ideal $I_{w}$ of
$C\\{\mathbf{a}^{+},\mathbf{a}^{-},\mathbf{a}^{0},\mathbf{b}_{w}\\}$ is prime.
Indeed, we gauge transform by root group elements belonging to negative roots.
Let $u_{-\alpha_{i_{j}}}(b_{i_{j}})$ be one of them and let $c$ be the
coefficient of some basis element of height $h$ in the linear representation
of

(6) $u_{-\alpha_{i_{j-1}}}(b_{i_{j-1}})\dots
u_{-\alpha_{i_{1}}}(b_{i_{1}}).A\,.$

By Remark 3.1 the effect of the adjoint action on $c$ is that we potentially
add to $c$ a monomial in $b_{i_{j}}$ and other coefficients of (6) which
belong to basis elements for roots of height greater than $h$. Further, Remark
3.2 implies that the effect of the logarithmic derivative of
$u_{-\alpha_{i_{j}}}(b_{i_{j}})$ is that we add the derivative of $b_{i_{j}}$
to the coefficient of the basis element $X_{-\alpha_{i_{j}}}$ in the linear
representation of (6). Altogether the effect of gauge transforming (6) with
$u_{-\alpha_{i_{j}}}(b_{i_{j}})$ is that we add to the coefficient $c$
differential monomials in the indeterminate $b_{i_{j}}$ and in those
indeterminates $\mathbf{a}^{+},\mathbf{a}^{-},\mathbf{a}^{0}$ which are of
height greater than $h$. We conclude inductively that for each basis element
$X_{\alpha_{i}}$, $X_{-\alpha_{i}}$ or $H_{i}$ of $\mathfrak{g}$, the
differential indeterminate which is the corresponding coefficient in (4)
appears in the coefficient of the same basis element in

(7) $u_{w}(\mathbf{b}_{w}).A$

only linearly with constant coefficient. Finally gauge transforming (7) with
$n(w)$ is only a permutation of the entries and so $f^{+}_{w,l+1}$, …,
$f^{+}_{w,m}$ can be solved for pairwise distinct differential indeterminates.

######  Lemma 5.1.

The differential ideal $I_{w}$ is prime.

We compute a Thomas decomposition $S_{w}^{1}$, …, $S_{w}^{r}$ of $S_{w}$ with
respect to an elimination ranking satisfying

$\\{\mathbf{b}_{w}\\}\gg\\{\mathbf{a}^{+},\,\mathbf{a}^{-},\,\mathbf{a}^{0}\,\\}\,.$

For $i\in\\{1,\ldots,r\\}$ let $I_{w}^{i}$ be the differential ideal of
$C\\{\mathbf{a}^{+},\mathbf{a}^{-},\mathbf{a}^{0},\mathbf{b}_{w}\\}$ generated
by the left hand sides of the equations in $S_{w}^{i}$, and define $q_{w}^{i}$
as the product of the initials and separants of all these generators. Then, by
Proposition 4.5, we have

$I_{w}=\left(I_{w}^{1}:(q_{w}^{1})^{\infty}\right)\cap\ldots\cap\left(I_{w}^{r}:(q_{w}^{r})^{\infty}\right).$

Since $I_{w}$ is prime, there exists a unique $j\in\\{1,\ldots,r\\}$ such that
$I_{w}=I_{w}^{j}:(q_{w}^{j})^{\infty}$. Hence, the simple system $S_{w}^{j}$
is generic in the sense that the solution set of $S_{w}^{j}$ is a dense subset
of the solution set of $S_{w}$ (cf. also [Rob14, Subsect. 2.2.3]).

Let $\mathcal{T}_{w}$ be the subset of $S_{w}^{j}$ consisting of the equations
and inequations which only involve $\mathbf{a}^{+}$, $\mathbf{a}^{-}$,
$\mathbf{a}^{0}$ and their derivatives. Let $E_{w}$ be the differential ideal
of $C\\{\mathbf{a}^{+},\mathbf{a}^{-},\mathbf{a}^{0}\\}$ which is generated by
the left hand sides of the equations $\mathcal{T}_{w}$, and define $q$ as the
product of the initials and separants of these generators. Because of the
choice of elimination ranking we have

$\left(I_{w}^{j}:(q_{w}^{j})^{\infty}\right)\cap
C\\{\mathbf{a}^{+},\mathbf{a}^{-},\mathbf{a}^{0}\\}\,=\,E_{w}:q^{\infty}.$

In principle, this approach allows to decide effectively whether gauge
transformation by an element of a Bruhat cell defined by $w$ into normal form
is possible by checking whether $E_{w}:q^{\infty}$ is the zero ideal.
Moreover, by trying the finitely many elements of $\mathcal{W}$, a feasible
one for gauge transformation to normal form can be found if one exists. In
fact, we are going to predict such a feasible element and hence prove that
gauge transformation to normal form is possible.

######  Definition 5.2.

For $w\in\mathcal{W}$ we consider the two conditions:

(A) $\exists\,\Psi\subseteq\Phi^{-}\mbox{ such that
}w(\Psi)=\Phi^{+}\setminus\Delta$

and

(B) $w(\Phi^{+})\cap\Phi^{-}\supseteq\Psi\,.$

An element $w\in\mathcal{W}$ is called _resolving_ if it satisfies Conditions
(A) and (B).

If $\Psi$ as above exists, then it is uniquely determined by $w$, namely as

$\Psi=w^{-1}(\Phi^{+}\setminus\Delta).$

In particular, we have $|\Psi|=m-l$.

######  Proposition 5.3.

For every classical group $G$ the associated Weyl group $\mathcal{W}$ contains
a resolving element.

###### Proof.

Since the root system of a classical group is irreducible, there is a unique
Weyl group element $\overline{w}\in\mathcal{W}$ which maps $\Phi^{+}$ to
$\Phi^{-}$. As a consequence we have $\overline{w}(\Delta)=\Delta^{-}$ and
because $\overline{w}$ has order two, it follows that
$\overline{w}(\Phi^{-})=\Phi^{+}$ and $\overline{w}(\Delta^{-})=\Delta$.
Define $\Psi:=\Phi^{-}\setminus\Delta^{-}$. One easily checks that $\Psi$ and
$\overline{w}$ satisfy Condition (A) and Condition (B). ∎

######  Definition 5.4.

A ranking $>$ on
$C\\{\mathbf{a}^{+},\mathbf{a}^{-},\mathbf{a}^{0},\mathbf{b}\\}$ is said to be
_adapted to the root system_ if the following three conditions are satisfied.

  1. (a)

The ranking $>$ is an elimination ranking satisfying
$\\{\mathbf{b}\\}\gg\\{\mathbf{a}^{+},\,\mathbf{a}^{-},\,\mathbf{a}^{0}\\}$.

  2. (b)

The restriction of $>$ to $C\\{\mathbf{b}\\}$ is an orderly ranking.

  3. (c)

For all $i$, $j\in\\{1,\ldots,m\\}$ we have

$b_{i}>b_{j}\quad\Rightarrow\quad|\operatorname{ht}(-\alpha_{i})|\geq|\operatorname{ht}(-\alpha_{j})|\,.$

Obviously a ranking that is adapted to the root system exists.

######  Theorem 5.5.

Let $w\in\mathcal{W}$ be resolving and assume that the ranking $>$ is adapted
to the root system. Then we have:

  1. (a)

The differential system

$S_{w}:\quad f^{+}_{w,l+1}=0,\quad\dots,\quad f^{+}_{w,m}=0$

is simple with respect to $>$.

  2. (b)

For $r\in\\{l+1,\dots,m\\}$ the leader of $f^{+}_{w,r}$ is $b_{i}^{\prime}$
corresponding to $-\alpha_{i}=w^{-1}(\alpha_{r})\in\Psi$.

  3. (c)

The differential polynomials $f^{+}_{w,l+1}$, …,$f^{+}_{w,m}$ are linear in
their leaders and have constant initials.

  4. (d)

The differential ideal generated by $f^{+}_{w,l+1}$, …,$f^{+}_{w,m}$ is prime.

  5. (e)

If the leader of $f^{+}_{w,r}$ is $b_{i}^{\prime}$, then $f^{+}_{w,r}$
contains the variable $a^{-}_{i}$ linearly with constant coefficient and
variables $a^{-}_{j}$ with
$|\operatorname{ht}(-\alpha_{j})|<|\operatorname{ht}(-\alpha_{i})|$ do not
appear in $f^{+}_{w,r}$.

###### Proof.

To begin with we compute the logarithmic derivative of

$u(\mathbf{b})\,=\,u_{-\alpha_{1}}(b_{1})\cdots u_{-\alpha_{m}}(b_{m}).$

We shall argue that we obtain

$\ell\delta(u(\mathbf{b}))=\sum_{i=1}^{m}\operatorname{Ad}(u_{-\alpha_{1}}(b_{1})\ldots
u_{-\alpha_{i-1}}(b_{i-1}))\,\big{(}\ell\delta(u_{-\alpha_{i}}(b_{i}))\big{)}=\sum_{i=1}^{m}(b_{i}^{\prime}+p_{i})X_{-\alpha_{i}}\,,$

where $p_{i}$ is a differential polynomial in $\mathbf{b}$ with
$\mathrm{ld}(b_{i}^{\prime}+p_{i})=b_{i}^{\prime}$. By Remark 3.2 we have
$\ell\delta(u_{-\alpha_{i}}(b_{i}))=b_{i}^{\prime}X_{-\alpha_{i}}$. By Remark
3.1 additional terms in the coefficient of $X_{-\alpha_{i}}$ can only stem
from the adjoint action

$\begin{array}[]{l}\operatorname{Ad}(u_{-\alpha_{1}}(b_{1})\ldots
u_{-\alpha_{j-1}}(b_{j-1}))\,\big{(}\ell\delta(u_{-\alpha_{j}}(b_{j}))\big{)}\\\\[5.0pt]
\qquad\qquad=\,\operatorname{Ad}(u_{-\alpha_{1}}(b_{1})\ldots
u_{-\alpha_{j-1}}(b_{j-1}))\,\big{(}b_{j}^{\prime}X_{-\alpha_{j}}\big{)}\end{array}$

such that $|\operatorname{ht}(-\alpha_{i})|>|\operatorname{ht}(-\alpha_{j})|$.
We conclude that every term in each $p_{i}$ is of degree at least two and is
divisible by precisely one derivative $b_{j}^{\prime}$ for some $j$ with
$|\operatorname{ht}(-\alpha_{i})|>|\operatorname{ht}(-\alpha_{j})|$. We obtain
now $u_{w}(\mathbf{b}_{w})$ from $u(\mathbf{b})$ by setting precisely those
$b_{k}$ equal to zero for which $-\alpha_{k}\not\in
w(\Phi^{+})\cap\Phi^{-}=\Psi$ and keeping the other $b_{i}$ as indeterminates.
Hence, for each $-\alpha_{i}\in\Psi$, the derivative $b_{i}^{\prime}$ appears
linearly and with constant coefficient in the coefficient of
$X_{-\alpha_{i}}$. Further, since the ranking is adapted to the root system,
the variable $b_{i}^{\prime}$ is the leader of that coefficient.

Adding the contribution $\operatorname{Ad}(u_{w}(\mathbf{b}_{w}))(A)$ of the
adjoint action to the logarithmic derivative yields the gauge transformation
of $A$ by $u_{w}(\mathbf{b}_{w})$. Since the adjoint action only creates
polynomial entries, we have

$\begin{array}[]{rcl}\operatorname{Ad}(u_{w}(\mathbf{b}_{w}))(A)+\ell\delta(u_{w}(\mathbf{b}_{w}))&=&\displaystyle\sum_{-\alpha_{i}\in\Psi}(b_{i}^{\prime}+p_{i}+q_{i})X_{-\alpha_{i}}\,+\\\\[5.0pt]
\lx@intercol\hfil\displaystyle\sum_{-\alpha_{i}\in\Phi\setminus\Psi}g_{-\alpha_{i}}X_{-\alpha_{i}}+\sum_{\alpha_{i}\in\Phi^{+}}g_{\alpha_{i}}X_{\alpha_{i}}+\sum_{i=1}^{l}g_{0,i}H_{i}\,,\hfil\lx@intercol\end{array}$

where $q_{i}$ as well as $g_{\alpha_{i}}$ and $g_{0,i}$ are polynomials in the
indeterminates $\mathbf{a}^{+}$, $\mathbf{a}^{-}$, $\mathbf{a}^{0}$,
$\mathbf{b}_{w}$ and where $g_{-\alpha_{i}}$ are differential polynomials in
the same indeterminates. By the first formula in Remark 3.1 the polynomial
$q_{i}$ contains the term $a^{-}_{i}$ linearly and with constant coefficient
and all other terms are non-constants and lie in the ring

$C[\mathbf{a}^{+},\mathbf{a}^{0},\mathbf{b}_{w},a^{-}_{j}\mid|\operatorname{ht}(-\alpha_{i})|>|\operatorname{ht}(-\alpha_{j})|].$

Again, because the ranking is adapted to the root system, the leader in the
coefficient of $X_{-\alpha_{i}}$ with $-\alpha_{i}\in\Psi$ is still
$b_{i}^{\prime}$. By Condition (A), after a further gauge transformation by
$n(w)$, these coefficients become the left hand sides in

$S_{w}:\quad f^{+}_{w,l+1}=0,\quad\dots,\quad f^{+}_{w,m}=0.$

It follows that they satisfy statements (b), (c) and (e). Since $S_{w}$
consists of equations with pairwise distinct leaders and the left hand sides
are linear in their leaders with constant initials, Condition (a) of
Definition 4.2 is satisfied. Moreover, since the leaders of these equations
involve pairwise distinct differential indeterminates, Condition (b) of
Definition 4.2 holds as well. Since $S_{w}$ contains no inequations, we
conclude that the differential system $S_{w}$ is simple with respect to $>$.
Finally, statement (d) of the theorem follows from Lemma 5.1. ∎

######  Remark 5.6.

There may be Weyl group elements $w\in\mathcal{W}$ which are not resolving,
but which lead to a consistent system

$S_{w}:\quad f^{+}_{w,l+1}=0,\quad\dots,\quad f^{+}_{w,m}=0.$

For example, when $G=\mathrm{SP}_{8}$, i.e. the group of type $C_{4}$, in the
representation defined in [Bou75, VIII, §13.3], the Weyl group element

${\small w=\left(\begin{array}[]{cccccccc}0&0&0&0&0&1&0&0\\\
0&0&0&0&0&0&0&-1\\\ 0&0&0&0&0&0&-1&0\\\ 0&0&0&-1&0&0&0&0\\\
0&0&0&0&-1&0&0&0\\\ 0&1&0&0&0&0&0&0\\\ 1&0&0&0&0&0&0&0\\\
0&0&-1&0&0&0&0&0\end{array}\right)}$

is not resolving, but one can check by computing a Thomas decomposition that
the corresponding system $S_{w}$ is consistent, i.e. admits solutions.

##  6\. A Normal Form for a Generic Equation

For a classical group $G$ with root system $\Phi$ and $l$ differential
indeterminates $\boldsymbol{t}=(t_{1},\dots,t_{l})$ over $C$ we constructed in
[Seib] and [Seia] the matrix $A_{G}(\boldsymbol{t})$ which defines a Picard-
Vessiot extension of $C\langle\boldsymbol{t}\rangle$ with differential Galois
group $G(C)$. For its definition let $A_{0}^{+}$ and $A_{0}^{-}$ be the sum of
basis elements of the root spaces corresponding to the simple roots and the
negatives of the simple roots, respectively. It is shown in [Seib, Lemma 6.4]
that there are $\gamma_{1},\dots,\gamma_{l}\in\Phi^{+}$ such that the sum

$\mathfrak{b}^{+}\,=\,\operatorname{ad}(A_{0}^{-})(\mathfrak{u}^{+})+\sum_{i=1}^{l}\mathfrak{g}_{\gamma_{i}}$

is direct and $\operatorname{ht}(\gamma_{i})=m_{i}$, where $m_{i}$ are the
exponents of $\mathfrak{g}$. We refer to $\gamma_{1},\dots,\gamma_{l}$ as
_complementary roots_.

######  Definition 6.1.

The matrix

$A_{G}(\boldsymbol{t})\,=\,A_{0}^{+}+\sum_{i=1}^{l}t_{i}X_{-\gamma_{i}}$

is called the _normal form matrix_ for $G$ (with respect to the complementary
roots $\gamma_{1},\dots,\gamma_{l}$).

Now let $A$ be as in Section 5. From [Jua04] it is known that $A$ defines a
Picard-Vessiot extension of
$C\langle\mathbf{a}^{+},\mathbf{a}^{-},\mathbf{a}^{0}\rangle$ with
differential Galois group $G(C)$. In this section we construct a differential
field extension

$C\langle\mathbf{a}^{+},\mathbf{a}^{-},\mathbf{a}^{0}\rangle\hookrightarrow\mathcal{L}$

such that the following conditions are satisfied:

  1. (a)

The field of constants of $\mathcal{L}$ is $C$.

  2. (b)

For the image $A\in\mathfrak{g}(\mathcal{L})$ and for some
$\boldsymbol{\overline{t}}\in\mathcal{L}^{l}$, the matrix $A$ is gauge
equivalent to $A_{G}(\boldsymbol{\overline{t}})$ over $\mathcal{L}$.

  3. (c)

The Galois group of the equation defined by $A_{G}(\boldsymbol{\overline{t}})$
over $\mathcal{L}$ is $G(C)$.

Let $w$ be a resolving element of the Weyl group $\mathcal{W}$ of $G$ and let
$>$ be a ranking on
$C\\{\mathbf{a}^{+},\mathbf{a}^{-},\mathbf{a}^{0},\mathbf{b}_{w}\\}$ which is
adapted to the root system. Then, by Theorem 5.5, the differential ideal
$I_{w}$ of
$C\\{\mathbf{a}^{+},\mathbf{a}^{-},\mathbf{a}^{0},\mathbf{b}_{w}\\}$ which is
generated by $f^{+}_{w,l+1}$, …, $f^{+}_{w,m}$ is prime. Furthermore, the
generators form a simple differential system and their leaders are the
$b_{i}^{\prime}$ corresponding to the elements $-\alpha_{i}$ of $\Psi$.
Moreover, the differential indeterminate $a^{-}_{i}$ occurs linearly with
constant coefficient in the generator with leader $b_{i}^{\prime}$ and is the
one with largest $|\operatorname{ht}(-\alpha_{j})|$ among the differential
indeterminates $a^{-}_{j}$ occurring in this generator. Since $a^{-}_{1}$, …,
$a^{-}_{m}$ are differentially algebraically independent over $C$, we have

(8) $I_{w}\cap
C\\{\mathbf{a}^{+},a^{-}_{i},\mathbf{a}^{0},\mathbf{b}_{w}\mid-\alpha_{i}\not\in\Psi,i=1,\ldots,m\\}\,=\,\\{0\\}.$

The kernel of the natural homomorphism

$\nu\colon
C\\{\mathbf{a}^{+},\mathbf{a}^{-},\mathbf{a}^{0},\mathbf{b}_{w}\\}\longrightarrow\mathcal{K}:=\operatorname{Frac}(C\\{\mathbf{a}^{+},\mathbf{a}^{-},\mathbf{a}^{0},\mathbf{b}_{w}\\}/I_{w})$

is equal to $I_{w}$. We denote the images of the indeterminates by

$\overline{b}_{i}:=\nu(b_{i}),\quad\overline{a}^{0}_{i}:=\nu(a^{0}_{i}),\quad\overline{a}^{+}_{i}:=\nu(a^{+}_{i}),\quad\overline{a}^{-}_{i}:=\nu(a^{-}_{i})$

and we write
$\mathbf{\overline{a}}^{0}=(\overline{a}^{0}_{1},\dots,\overline{a}^{0}_{l})$,
$\mathbf{\overline{a}}^{+}=(\overline{a}^{+}_{1},\dots,\overline{a}^{+}_{m})$,
$\mathbf{\overline{a}}^{-}=(\overline{a}^{-}_{1},\dots,\overline{a}^{-}_{m})$.
Moreover, denote by $\mathbf{a}_{\neg\Psi}^{-}$ and
$\mathbf{\overline{a}}_{\neg\Psi}^{-}$ the tuples of the $l$ elements
$a^{-}_{i}$ and $\overline{a}^{-}_{i}$, respectively, with the property that
$-\alpha_{i}\notin\Psi$. Since

(9) $I_{w}\cap
C\\{\mathbf{a}^{-},\mathbf{a}^{+},\mathbf{a}^{0}\\}\,=\,\\{0\\},$

the elements $\mathbf{\overline{a}}^{0}$, $\mathbf{\overline{a}}^{+}$ and
$\mathbf{\overline{a}}^{-}$ are differentially algebraically independent over
$C$. We can identify the differential field
$C\langle\mathbf{a}^{-},\mathbf{a}^{+},\mathbf{a}^{0}\rangle$ with the
differential subfield
$C\langle\mathbf{\overline{a}}^{-},\mathbf{\overline{a}}^{+},\mathbf{\overline{a}}^{0}\rangle$
of $\mathcal{K}$, i.e., $\mathcal{K}$ is a differential extension field of
$C\langle\mathbf{\overline{a}}^{-},\mathbf{\overline{a}}^{+},\mathbf{\overline{a}}^{0}\rangle$
which is a purely differentially transcendental differential field over $C$.
We consider the differential subfield
$C\langle\mathbf{\overline{b}}_{w}\rangle$ of $\mathcal{K}$, where
$\mathbf{\overline{b}}_{w}=(\overline{b}_{i_{1}},\dots,\overline{b}_{i_{m-l}})$
and $\Psi=\\{-\alpha_{i_{1}},\dots,-\alpha_{i_{m-l}}\\}$.

######  Lemma 6.2.

The subset
$\\{\mathbf{\overline{a}}^{0},\mathbf{\overline{a}}^{+},\mathbf{\overline{a}}_{\neg\Psi}^{-}\\}$
of $\mathcal{K}$ is differentially algebraically independent over
$C\langle\mathbf{\overline{b}}_{w}\rangle$ and

$C\langle\mathbf{\overline{b}}_{w}\rangle\langle\mathbf{\overline{a}}^{0},\mathbf{\overline{a}}^{+},\mathbf{\overline{a}}_{\neg\Psi}^{-}\rangle\,=\,\mathcal{K}.$

###### Proof.

Let $P(\mathbf{Z})\in
C\langle\mathbf{\overline{b}}_{w}\rangle\\{\mathbf{Z}\\}$ be a differential
polynomial in the differential indeterminates
$\mathbf{Z}=(Z_{1},\dots,Z_{m+2l})$ such that
$P(\mathbf{\overline{a}}^{0},\mathbf{\overline{a}}^{+},\mathbf{\overline{a}}_{\neg\Psi}^{-})=0$.
After multiplying $P(\mathbf{Z})$ by a non-zero element of
$C\langle\mathbf{\overline{b}}_{w}\rangle$ we may assume that $P(\mathbf{Z})$
lies in the image of the composition of homomorphisms

$C\\{\mathbf{b}_{w}\\}\\{\mathbf{Z}\\}\longrightarrow
C\\{\mathbf{b}_{w},\mathbf{a}^{0},\mathbf{a}^{+},\mathbf{a}^{-}\\}\\{\mathbf{Z}\\}\longrightarrow\mathcal{K}\\{\mathbf{Z}\\},$

where the first homomorphism is induced by the inclusion

$C\\{\mathbf{b}_{w}\\}\longrightarrow
C\\{\mathbf{b}_{w},\mathbf{a}^{0},\mathbf{a}^{+},\mathbf{a}^{-}\\}$

and the second homomorphism is defined by applying $\nu$ to each coefficient
of a differential polynomial. Let $\widehat{P}(\mathbf{Z})\in
C\\{\mathbf{b}_{w}\\}\\{\mathbf{Z}\\}$ be any preimage of $P(\mathbf{Z})$
under the above composition of homomorphisms. Note that by definition of
$\mathbf{\overline{b}}_{w}$ and $\mathbf{\overline{a}}^{0}$,
$\mathbf{\overline{a}}^{+}$, $\mathbf{\overline{a}}^{-}$ the diagram

${C\\{\mathbf{b}_{w}\\}\\{\mathbf{Z}\\}}$${C\langle\mathbf{\overline{b}}_{w}\rangle\\{\mathbf{Z}\\}}$${\mathcal{K}}$$\scriptstyle{b_{i}\,\mapsto\,\overline{b}_{i}}$$\scriptstyle{\widehat{P}(\mathbf{Z})\,\mapsto\,\nu(\widehat{P}(\mathbf{a}^{0},\mathbf{a}^{+},\mathbf{a}_{\neg\Psi}^{-}))}$$\scriptstyle{P(\mathbf{Z})\,\mapsto\,P(\mathbf{\overline{a}}^{0},\mathbf{\overline{a}}^{+},\mathbf{\overline{a}}_{\neg\Psi}^{-})}$

is commutative. Hence,
$P(\mathbf{\overline{a}}^{0},\mathbf{\overline{a}}^{+},\mathbf{\overline{a}}_{\neg\Psi}^{-})=0$
implies
$\widehat{P}(\mathbf{a}^{0},\mathbf{a}^{+},\mathbf{a}_{\neg\Psi}^{-})\in\ker(\nu)=I_{w}$.
Now, since

$\widehat{P}(\mathbf{a}^{0},\mathbf{a}^{+},\mathbf{a}_{\neg\Psi}^{-})\in
I_{w}\cap
C\\{\mathbf{b}_{w},\mathbf{a}^{0},\mathbf{a}^{+},\mathbf{a}_{\neg\Psi}^{-}\\},$

we conclude with (8) that $\widehat{P}(\mathbf{Z})$ is the zero polynomial and
so is $P(\mathbf{Z})$.

The differential field $\mathcal{K}$ is generated by
$\mathbf{\overline{b}}_{w}$ and $\mathbf{\overline{a}}^{0}$,
$\mathbf{\overline{a}}^{+}$ and $\mathbf{\overline{a}}^{-}$ over $C$. We prove
now that $\overline{a}^{-}_{i}$ lies in
$C\langle\mathbf{\overline{b}}_{w}\rangle\langle\mathbf{\overline{a}}^{0},\mathbf{\overline{a}}^{+},\mathbf{\overline{a}}_{\neg\Psi}^{-}\rangle$
for every $-\alpha_{i}\in\Psi$, i.e., $\mathcal{K}$ is generated by the above
elements with $\mathbf{\overline{a}}_{\neg\Psi}^{-}$ instead of
$\mathbf{\overline{a}}^{-}$. We fix some $-\alpha_{i}\in\Psi$. Using
$f^{+}_{w,l+1},\dots,f^{+}_{w,m}$ and their properties stated in Theorem 5.5
we can express $\overline{a}^{-}_{i}$ in terms of $\mathbf{\overline{b}}_{w}$,
$\mathbf{\overline{a}}^{0}$, $\mathbf{\overline{a}}^{+}$,
$\mathbf{\overline{a}}_{\neg\Psi}^{-}$ and $\overline{a}^{-}_{j}$ with
$-\alpha_{j}\in\Psi$ of height greater than the height of $-\alpha_{i}$,
proceeding recursively and starting with the elements of greatest height. ∎

######  Lemma 6.3.

The field of constants of $\mathcal{K}$ is $C$.

###### Proof.

By Lemma 6.2 we have
$\mathcal{K}=C\langle\mathbf{\overline{b}}_{w},\mathbf{\overline{a}}^{0},\mathbf{\overline{a}}^{+},\mathbf{\overline{a}}_{\neg\Psi}^{-}\rangle$.
Then the field of constants of $\mathcal{K}$ is $C$, because
$\mathbf{\overline{b}}_{w}$, $\mathbf{\overline{a}}^{0}$,
$\mathbf{\overline{a}}^{+}$, $\mathbf{\overline{a}}_{\neg\Psi}^{-}$ are
differentially algebraically independent over $C$. Indeed, if $P(\mathbf{Z})$
is a differential polynomial over $C$ in the indeterminates
$\mathbf{Z}=(Z_{1},\ldots,Z_{2m+l})$ such that
$P(\mathbf{\overline{b}}_{w},\mathbf{\overline{a}}^{0},\mathbf{\overline{a}}^{+},\mathbf{\overline{a}}_{\neg\Psi}^{-})=0$,
then
$P(\mathbf{b}_{w},\mathbf{a}^{0},\mathbf{a}^{+},\mathbf{a}_{\neg\Psi}^{-})$
can be considered as an element of
$C\\{\mathbf{b}_{w},\mathbf{a}^{0},\mathbf{a}^{+},\mathbf{a}^{-}\\}$ which is
in the kernel of the natural homomorphism $\nu$. But (8) implies that
$P(\mathbf{Z})$ is the zero polynomial. ∎

The gauge transformation of
$A(\mathbf{\overline{a}}^{+},\mathbf{\overline{a}}^{-},\mathbf{\overline{a}}^{0})$
to a matrix $A_{G}(\mathbf{\overline{t}})$ in normal form with
$\mathbf{\overline{t}}=(\overline{t},\dots,\overline{t}_{l})$ in some finite
algebraic extension of $\mathcal{K}$ is done in three steps. First we show
that
$A(\mathbf{\overline{a}}^{+},\mathbf{\overline{a}}^{-},\mathbf{\overline{a}}^{0})$
is gauge equivalent to a matrix
$A(\mathbf{\overline{h}}^{+},\mathbf{\overline{h}}^{-},\mathbf{\overline{h}}^{0})$
with
$\mathbf{\overline{h}}^{+}=(\overline{h}_{1}^{+},\dots,\overline{h}^{+}_{l},0,\dots,0)$
over $\mathcal{K}$ and $\overline{h}^{+}_{1},\dots,\overline{h}^{+}_{l}$,
$\mathbf{\overline{h}}^{-}$, $\mathbf{\overline{h}}^{0}$ are differentially
algebraically independent over $C\langle\mathbf{\overline{b}}\rangle$ and
$C\langle\mathbf{\overline{b}}\rangle\langle\overline{h}^{+}_{1},\dots,\overline{h}^{+}_{l},\mathbf{\overline{h}}^{-},\mathbf{\overline{h}}^{0}\rangle=\mathcal{K}$.
In a second step we prove that
$A(\mathbf{\overline{h}}^{+},\mathbf{\overline{h}}^{-},\mathbf{\overline{h}}^{0})$
is gauge equivalent to a matrix
$A(\mathbf{\overline{g}}^{+},\mathbf{\overline{g}}^{-},\mathbf{\overline{g}}^{0})$
with $\mathbf{\overline{g}}^{+}=(1,\dots,1,0,\dots,0)$ with $l$ entries equal
to $1$ and $m-l$ entries equal to $0$, over an algebraic extension
$\mathcal{L}$ of $\mathcal{K}$. It will turn out that there is a differential
subfield $\mathcal{L}_{1}$ of $\mathcal{L}$ such that
$\mathbf{\overline{g}}^{-},\mathbf{\overline{g}}^{0}$ are differentially
algebraically independent over $\mathcal{L}_{1}$ and
$\mathcal{L}_{1}\langle\mathbf{\overline{g}}^{-},\mathbf{\overline{g}}^{0}\rangle=\mathcal{L}$.
In a last step we gauge transform
$A(\mathbf{\overline{g}}^{+},\mathbf{\overline{g}}^{-},\mathbf{\overline{g}}^{0})$
to a matrix of shape $A_{G}(\mathbf{\overline{t}})$ and we prove that there is
a differential subfield $\mathcal{L}_{2}$ such that
$\mathcal{L}_{2}\langle\mathbf{\overline{t}}\rangle=\mathcal{L}$ and
$\mathbf{\overline{t}}$ is differentially algebraically independent over
$\mathcal{L}_{2}$. Applying Theorem 7.4 we obtain that the differential Galois
group of $A_{G}(\mathbf{\overline{t}})$ over $\mathcal{L}$ is the full group
$G$ and so is the one for
$A(\mathbf{\overline{a}}^{+},\mathbf{\overline{a}}^{-},\mathbf{\overline{a}}^{0})$
over $\mathcal{L}$.

Extending the notation of (5) let

$\boldsymbol{u}(\mathbf{\overline{b}}_{w})\,=\,u_{-\alpha_{i_{1}}}(\overline{b}_{i_{1}})\cdots
u_{-\alpha_{i_{m-l}}}(\overline{b}_{i_{m-l}})$

with the previously fixed resolving Weyl group element $w$.

######  Lemma 6.4.

Gauge transformation yields

$(n(w)\boldsymbol{u}(\mathbf{\overline{b}}_{w})).A(\mathbf{\overline{a}}^{+},\mathbf{\overline{a}}^{-},\mathbf{\overline{a}}^{0})\,=\,A(\mathbf{\overline{h}}^{+},\mathbf{\overline{h}}^{-},\mathbf{\overline{h}}^{0})$

with
$\mathbf{\overline{h}}^{+}=(\overline{h}^{+}_{1},\dots,\overline{h}^{+}_{l},0,\dots,0)\in\mathcal{K}^{m}$,
$\mathbf{\overline{h}}^{-}\in\mathcal{K}^{m}$ and
$\mathbf{\overline{h}}^{0}\in\mathcal{K}^{l}$. Further we have

$C\langle\mathbf{\overline{b}}_{w}\rangle\langle\mathbf{\overline{h}}^{+},\mathbf{\overline{h}}^{-},\mathbf{\overline{h}}^{0}\rangle\,=\,\mathcal{K}$

and $\overline{h}^{+}_{1},\dots,\overline{h}^{+}_{l}$,
$\mathbf{\overline{h}}^{-}$, $\mathbf{\overline{h}}^{0}$ are differentially
algebraically independent over $C\langle\mathbf{\overline{b}}_{w}\rangle$.

###### Proof.

First we apply gauge transformation by
$\boldsymbol{u}(\mathbf{\overline{b}}_{w})$ to
$A(\mathbf{\overline{a}}^{+},\mathbf{\overline{a}}^{-},\mathbf{\overline{a}}^{0})$
and obtain

$\boldsymbol{u}(\mathbf{\overline{b}}_{w}).A(\mathbf{\overline{a}}^{+},\mathbf{\overline{a}}^{-},\mathbf{\overline{a}}^{0})\,=\,\sum_{i=1}^{m}\overline{p}^{+}_{i}X_{\alpha_{i}}+\sum_{i=1}^{m}\overline{p}^{-}_{i}X_{-\alpha_{i}}+\sum_{j=1}^{l}\overline{p}^{0}_{j}H_{j}\,,$

where $\overline{p}^{\pm}_{i}$, $\overline{p}^{0}_{j}\in
C\\{\mathbf{\overline{b}}_{w}\\}\\{\mathbf{\overline{a}}^{+},\mathbf{\overline{a}}^{0},\mathbf{\overline{a}}^{-}\\}\subset\mathcal{K}$.
First of all, by construction of the extension field $\mathcal{K}$ and its
elements $\mathbf{\overline{b}}_{w}$, the coefficients $\overline{p}^{-}_{i}$
with $-\alpha_{i}\in\Psi$ vanish. Let
$\mathbf{\overline{p}}^{+}=(\overline{p}^{+}_{1},\dots,\overline{p}^{+}_{m})$
and
$\mathbf{\overline{p}}^{0}=(\overline{p}^{0}_{1},\dots,\overline{p}^{0}_{m})$
and write $\mathbf{\overline{p}}^{-}_{\neg\Psi}$ for the $l$-tuple of those
$\overline{p}^{-}_{i}$ such that $-\alpha_{i}\notin\Psi$. By Lemma 6.2 we have
$\mathcal{K}=C\langle\mathbf{\overline{b}}_{w}\rangle\langle\mathbf{\overline{a}}^{0},\mathbf{\overline{a}}^{+},\mathbf{\overline{a}}_{\neg\Psi}^{-}\rangle$
and we are going to prove that

$C\langle\mathbf{\overline{b}}_{w}\rangle\langle\mathbf{\overline{a}}^{0},\mathbf{\overline{a}}^{+},\mathbf{\overline{a}}_{\neg\Psi}^{-}\rangle\,=\,C\langle\mathbf{\overline{b}}_{w}\rangle\langle\mathbf{\overline{p}}^{+},\mathbf{\overline{p}}^{0},\mathbf{\overline{p}}_{\neg\Psi}^{-}\rangle$

by showing that
$\mathbf{\overline{a}}^{0},\mathbf{\overline{a}}^{+},\mathbf{\overline{a}}_{\neg\Psi}^{-}$
are contained in
$C\langle\mathbf{\overline{b}}_{w}\rangle\langle\mathbf{\overline{p}}^{+},\mathbf{\overline{p}}^{0},\mathbf{\overline{p}}_{\neg\Psi}^{-}\rangle$.
Then we conclude that
$\mathbf{\overline{p}}^{+},\mathbf{\overline{p}}^{0},\mathbf{\overline{p}}_{\neg\Psi}^{-}$
are differentially algebraically independent over
$C\langle\mathbf{\overline{b}}_{w}\rangle$, since by Lemma 6.2 the same is
true for
$\mathbf{\overline{a}}^{+},\mathbf{\overline{a}}^{0},\mathbf{\overline{a}}_{\neg\Psi}^{-}$
(by the invariance of the differential transcendence degree).

The desired equality of the differential fields follows from the fact that one
can recursively solve the differential polynomials
$\mathbf{\overline{p}}^{+},\mathbf{\overline{p}}^{0},\mathbf{\overline{p}}_{\neg\Psi}^{-}$
for the indeterminates $\mathbf{\overline{a}}^{+}$,
$\mathbf{\overline{a}}^{0}_{i}$, $\mathbf{\overline{a}}_{\neg\Psi}^{-}$,
starting with the indeterminate corresponding to the root of greatest height.
More precisely, since we gauge transformed
$A(\mathbf{\overline{a}}^{+},\mathbf{\overline{a}}^{-},\mathbf{\overline{a}}^{0})$
with $\boldsymbol{u}(\mathbf{\overline{b}}_{w})$, i.e. a product of root group
elements corresponding to negative roots only, the differential polynomial
$p^{+}_{m}$, which belongs to the greatest positive root $\alpha_{m}$, is
simply the indeterminate $\overline{a}^{+}_{m}$. Hence, we trivially have
$\overline{a}^{+}_{m}\in
C\langle\mathbf{\overline{b}}_{w}\rangle\langle\mathbf{\overline{p}}^{+},\mathbf{\overline{p}}^{0},\mathbf{\overline{p}}_{\neg\Psi}^{-}\rangle$.
Let $\overline{p}_{i}$ be one of the coefficients corresponding to a root of
height less than $m$ and let $\overline{a}_{i}$ be the indeterminate among
$\mathbf{\overline{a}}^{0},\mathbf{\overline{a}}^{+},\mathbf{\overline{a}}_{\neg\Psi}^{-}$
corresponding to the same root. Then in $\overline{p}_{i}$ the indeterminate
$\overline{a}_{i}$ occurs linearly with non-zero constant coefficient.
Moreover, the indeterminates other than $\overline{a}_{i}$ among
$\mathbf{\overline{a}}^{+}$, $\mathbf{\overline{a}}^{0}$,
$\mathbf{\overline{a}}_{\neg\Psi}^{-}$ occurring in $\overline{p}_{i}$
correspond to roots of height neither smaller than nor equal to the one
corresponding to $\overline{a}_{i}$, where we consider the indeterminates
$\mathbf{\overline{a}}^{0}$ to correspond to height zero. So we recursively
obtain that $\mathbf{\overline{a}}^{+}$, $\mathbf{\overline{a}}^{0}$,
$\mathbf{\overline{a}}_{\neg\Psi}^{-}$ lie in
$C\langle\mathbf{\overline{b}}_{w}\rangle\langle\mathbf{\overline{p}}^{+},\mathbf{\overline{p}}^{0},\mathbf{\overline{p}}_{\neg\Psi}^{-}\rangle$
as claimed.

Finally the gauge transformation of the intermediate matrix with $n(w)$ yields
a matrix whose coefficients $\mathbf{\overline{h}}^{+}$,
$\mathbf{\overline{h}}^{-}$ and $\mathbf{\overline{h}}^{0}$ have the desired
properties. ∎

######  Lemma 6.5.

There exists a finite algebraic extension $\mathcal{L}$ of $\mathcal{K}$ and
an element $t$ of the torus $T(\mathcal{L})$ of $\mathcal{L}$-rational points
such that
$A(\mathbf{\overline{h}}^{+},\mathbf{\overline{h}}^{-},\mathbf{\overline{h}}^{0})$
is gauge equivalent by $t$ to a matrix of shape
$A(\mathbf{\overline{g}}^{+},\mathbf{\overline{g}}^{-},\mathbf{\overline{g}}^{0})$
with

$\mathbf{\overline{g}}^{+}\,=\,(\underbrace{1,\dots,1}_{l},\underbrace{0,\dots,0}_{m-l}),$

$\mathbf{\overline{g}}^{-}\in\mathcal{L}^{m}$ and
$\mathbf{\overline{g}}^{0}\in\mathcal{L}^{l}$. There is a differential
subfield $\mathcal{L}_{1}$ of $\mathcal{L}$ such that
$\mathcal{L}_{1}\langle\mathbf{\overline{g}}^{-},\mathbf{\overline{g}}^{0}\rangle=\mathcal{L}$
and $\mathbf{\overline{g}}^{-}$ and $\mathbf{\overline{g}}^{0}$ are
differentially algebraically independent over $\mathcal{L}_{1}$.

###### Proof.

For indeterminates $\boldsymbol{x}=(x_{1},\dots,x_{l})$ let

$\boldsymbol{t}(\boldsymbol{x})\,=\,t_{1}(x_{1})\cdots t_{l}(x_{l})\,,$

where for a simple root $\alpha_{i}$ the torus element $t_{i}(x_{i})$ is the
image of $\mathrm{diag}(x_{i},x_{i}^{-1})$ under the homomorphism from
$\mathrm{SL}_{2}$ to the group generated by $U_{\alpha_{i}}$ and
$U_{-\alpha_{i}}$. For two simple roots $\alpha_{i}$, $\alpha_{j}\in\Delta$ we
have

$\operatorname{Ad}(t_{i}(x_{i}))(X_{\alpha_{j}})\,=\,x_{i}^{\langle\alpha_{i},\alpha_{j}\rangle}X_{\alpha_{j}}\,,$

where $\langle\alpha_{i},\alpha_{j}\rangle$ is the integer at the entry
$(i,j)$ of the corresponding Cartan matrix and so the coefficient of
$X_{\alpha_{j}}$ in the linear representation of

$\operatorname{Ad}(\boldsymbol{t}(\boldsymbol{x}))(A(\mathbf{\overline{h}}^{+},\mathbf{\overline{h}}^{-},\mathbf{\overline{h}}^{0}))$

is $\overline{h}^{+}_{j}x_{1}^{\langle\alpha_{1},\alpha_{j}\rangle}\cdots
x_{l}^{\langle\alpha_{l},\alpha_{j}\rangle}$. We require now that

(10)
$\overline{h}^{+}_{1}\prod_{i=1}^{l}x_{i}^{\langle\alpha_{i},\alpha_{1}\rangle}\,=\,1,\quad\dots,\quad\,\overline{h}^{+}_{l}\prod_{i=1}^{l}x_{i}^{\langle\alpha_{i},\alpha_{l}\rangle}\,=\,1.$

By choosing a logarithm, solutions
$\boldsymbol{\overline{x}}=(\overline{x}_{1},\dots,\overline{x}_{l})$ of the
above equations are obtained as exponential functions of solutions to the
linear system

$\Gamma\,(\log(x_{1}),\ldots,\log(x_{l}))^{tr}\,\,=\,\,-(\log(\overline{h}^{+}_{1}),\ldots,\log(\overline{h}^{+}_{l}))^{tr}\,,$

where $\Gamma$ is the Cartan matrix of $\mathfrak{g}$. Hence, each
$\overline{x}_{i}$ is a certain product of powers of $\overline{h}^{+}_{1}$,
…, $\overline{h}^{+}_{l}$ with rational exponents $q_{i}$. Let
$\mathcal{L}=\mathcal{K}(\boldsymbol{\overline{x}})$ be the algebraic
extension of $\mathcal{K}$ generated by $\boldsymbol{\overline{x}}$ (whose
structure as a differential field is uniquely determined). For the algebraic
extension
$\mathcal{L}_{1}=C\langle\mathbf{\overline{b}}_{w}\rangle\langle\mathbf{\overline{h}}^{+}\rangle(\boldsymbol{\overline{x}})$
we have

$\mathcal{L}_{1}\langle\mathbf{\overline{h}}^{-},\mathbf{\overline{h}}^{0}\rangle\,=\,\mathcal{L}\,.$

Since $\mathbf{\overline{h}}^{-},\mathbf{\overline{h}}^{0}$ and
$\mathbf{\overline{h}}^{+}$ are differentially algebraically independent over
$C\langle\mathbf{\overline{b}}_{w}\rangle$ by Lemma 6.4 and since
$\mathcal{L}_{1}$ is defined over
$C\langle\mathbf{\overline{b}}_{w}\rangle\langle\mathbf{\overline{h}}^{+}\rangle$,
we conclude that $\mathbf{\overline{h}}^{-},\mathbf{\overline{h}}^{0}$ are
differentially algebraically independent over $\mathcal{L}_{1}$. Let
$\mathbf{\overline{g}}^{+},\mathbf{\overline{g}}^{-},\mathbf{\overline{g}}^{0}$
be the coefficients of the linear representation of

$t.A(\mathbf{\overline{h}}^{+},\mathbf{\overline{h}}^{-},\mathbf{\overline{h}}^{0})\,=\,A(\mathbf{\overline{g}}^{+},\mathbf{\overline{g}}^{-},\mathbf{\overline{g}}^{0}),$

where $t=\boldsymbol{t}(\boldsymbol{\overline{x}})$. As above, for
$\alpha\in\Phi$ the effect of the adjoint action of $t_{i}(\overline{x}_{i})$
on $X_{\alpha}$ is multiplication by the scalar
$\overline{x}_{i}^{\langle\alpha_{i},\alpha\rangle}$ and the action of
$\operatorname{Ad}(\boldsymbol{t}(\boldsymbol{\overline{x}}))$ on the Cartan
subalgebra is trivial. Further the logarithmic derivative of
$\boldsymbol{t}(\boldsymbol{\overline{x}})$ is an element of $\mathfrak{h}$.
We conclude that, since $\boldsymbol{\overline{x}}\in\mathcal{L}_{1}$, the
elements $\overline{g}^{0}_{i}$ and $\overline{h}^{0}_{i}$ differ by an
element of $\mathcal{L}_{1}$ and that $\overline{g}^{+}_{j}$,
$\overline{g}^{-}_{j}$ are obtained from $\overline{h}^{+}_{j}$,
$\overline{h}^{-}_{j}$ by multiplication by an element of $\mathcal{L}_{1}$.
Then, by the choice of $\boldsymbol{\overline{x}}$, the first $l$ entries of
$\mathbf{\overline{g}}^{+}$ are equal to $1$ and the others remain zero. We
conclude further that $\mathbf{\overline{h}}^{-}$, $\mathbf{\overline{h}}^{0}$
are contained in
$\mathcal{L}_{1}\langle\mathbf{\overline{g}}^{-},\mathbf{\overline{g}}^{0}\rangle$
and so we have

$\mathcal{L}_{1}\langle\mathbf{\overline{g}}^{-},\mathbf{\overline{g}}^{0}\rangle\,=\,\mathcal{L}.$

It follows that $\mathbf{\overline{g}}^{-},\mathbf{\overline{g}}^{0}$ are
differentially algebraically independent over $\mathcal{L}_{1}$, since the
same holds for $\mathbf{\overline{h}}^{-}$, $\mathbf{\overline{h}}^{0}$ over
$\mathcal{L}_{1}$ (due to the invariance of the differential transcendence
degree). ∎

######  Lemma 6.6.

There exists
$\mathbf{\overline{t}}=(\overline{t}_{1},\dots,\overline{t}_{l})\in\mathcal{L}_{1}\\{\mathbf{\overline{g}}^{-},\mathbf{\overline{g}}^{0}\\}^{l}$
with the following properties:

  1. (a)

The matrix
$A(\mathbf{\overline{g}}^{+},\mathbf{\overline{g}}^{-},\mathbf{\overline{g}}^{0})$
is gauge equivalent to $A_{G}(\mathbf{\overline{t}})$ over $\mathcal{L}$.

  2. (b)

Let $i_{1},\dots,i_{l}$ be the indices of the complementary roots
$-\alpha_{i_{1}},\dots,-\alpha_{i_{l}}$. Then, for $j=1,\dots,l$, the
differential polynomial $\overline{t}_{j}$ does not involve the indeterminates
$\overline{g}^{-}_{i_{j+1}},\dots,\overline{g}^{-}_{i_{l}}$ nor any proper
derivative of $\overline{g}^{-}_{i_{j}}$, but the degree of $\overline{t}_{j}$
in the indeterminate $\overline{g}^{-}_{i_{j}}$ is equal to one, and the
coefficient of $\overline{g}^{-}_{i_{j}}$ in $\overline{t}_{j}$ is constant.

  3. (c)

Let $\mathcal{L}_{2}$ be the differential field generated by
$\mathbf{\overline{g}}^{0}$ and

$\\{\overline{g}^{-}_{i}\in\mathbf{\overline{g}}^{-}\mid
i\notin\\{i_{1},\dots,i_{l}\\}\\}$

over $\mathcal{L}_{1}$. Then
$\mathcal{L}_{2}\langle\mathbf{\overline{t}}\rangle=\mathcal{L}$ and the
elements $\mathbf{\overline{t}}$ are differentially algebraically independent
over $\mathcal{L}_{2}$.

###### Proof.

It follows from [Seib, Lemma 6.8] (Transformation Lemma) that the matrix
$A(\mathbf{\overline{g}}^{+},\mathbf{\overline{g}}^{-},\mathbf{\overline{g}}^{0})$
is gauge equivalent by a unipotent matrix $u\in
U^{-}(\mathcal{L}_{1}\\{\mathbf{\overline{g}}^{-},\mathbf{\overline{g}}^{0}\\})$
to a matrix $A_{G}(\mathbf{\overline{t}})$ with
$\mathbf{\overline{t}}=(\overline{t}_{1},\dots,\overline{t}_{l})$ in
$\mathcal{L}_{1}\langle\mathbf{\overline{g}}^{-},\mathbf{\overline{g}}^{0}\rangle^{l}$.
More precisely, since $u$ is unipotent with entries in
$\mathcal{L}_{1}\\{\mathbf{\overline{g}}^{-},\mathbf{\overline{g}}^{0}\\}$,
the inverse of $u$ also has entries in
$\mathcal{L}_{1}\\{\mathbf{\overline{g}}^{-},\mathbf{\overline{g}}^{0}\\}$ and
so we have that
$\overline{t}_{i}\in\mathcal{L}_{1}\\{\mathbf{\overline{g}}^{-},\mathbf{\overline{g}}^{0}\\}$
for all $1\leq i\leq l$. This shows (a).

In the proof of the Transformation Lemma we gauge transform
$A(\mathbf{\overline{g}}^{+},\mathbf{\overline{g}}^{-},\mathbf{\overline{g}}^{0})$
successively with products of root group elements corresponding to roots of
height $-1,\dots,-m$. The effect on
$A(\mathbf{\overline{g}}^{+},\mathbf{\overline{g}}^{-},\mathbf{\overline{g}}^{0})$
is that we successively eliminate all entries corresponding to roots of height
$0,-1,-2,\dots,-(m-1)$ except the complementary roots. In each step the
parametrization of the product of root group elements of height $-k$ depends
linearly on the entries which we are going to eliminate, i.e. the entries
corresponding to the roots of height $-(k-1)$. The only non-zero entries in
$A(\mathbf{\overline{g}}^{+},\mathbf{\overline{g}}^{-},\mathbf{\overline{g}}^{0})$
which correspond to positive roots are the entries belonging to the simple
roots. In each step the effect of the adjoint action coming from the positive
simple roots is only the elimination described above and all other
contributions of the adjoint action stem from entries belonging to the
negative roots. One proves now inductively on $-k=-1,\dots,-m$ the following
claim: After performing step $-k$ an entry corresponding to a root of height
$h$ does not involve indeterminates among the $\mathbf{\overline{g}}^{-}$
corresponding to roots of height less than $h$ nor a proper derivative of
$\overline{g}^{-}_{h}$, but the degree of this entry in $\overline{g}^{-}_{h}$
is equal to one and the coefficient is constant. In each step of the induction
one concludes first with Remark 3.1 and 3.2 that to entries of height less
than or equal to $-k$ only differential polynomials in indeterminates
$\mathbf{\overline{g}}^{0}$ and those of $\mathbf{\overline{g}}^{-}$ which
correspond to roots of height greater than $-k$ are added. Secondly one
concludes with Remark 3.1 and 3.2 that to entries of height greater than $-k$
nothing is added. Since the complementary roots, which among themselves are
ordered by height, are the only ones which are not eliminated we conclude
statement (b).

Next we prove that
$\mathcal{L}_{2}\langle\mathbf{\overline{t}}\rangle=\mathcal{L}_{1}\langle\mathbf{\overline{g}}^{0},\mathbf{\overline{g}}^{-}\rangle=\mathcal{L}$.
We need to show that $\mathcal{L}_{2}\langle\mathbf{\overline{t}}\rangle$
contains all $\overline{g}^{-}_{i_{1}},\dots,\overline{g}^{-}_{i_{l}}$. By
statement (b) we can successively express the elements
$\overline{g}^{-}_{i_{1}},\dots,\overline{g}^{-}_{i_{l}}$ in terms of
$\mathbf{\overline{t}}$ and

$\\{\mathbf{\overline{g}}^{0},\mathbf{\overline{g}}^{-}\\}\setminus\\{\overline{g}^{-}_{i_{1}},\dots,\overline{g}^{-}_{i_{l}}\\}$

over $\mathcal{L}_{1}$. Indeed, $\overline{t}_{1}$ depends linearly on
$\overline{g}^{-}_{i_{1}}$ and does not involve any other indeterminate
$\overline{g}^{-}_{i_{2}},\dots,\overline{g}^{-}_{i_{l}}$ nor any proper
derivative of $\overline{g}^{-}_{i_{1}}$ and so we can express
$\overline{g}^{-}_{i_{1}}$ as stated. The expression $\overline{t}_{2}$ does
not involve any of the indeterminates
$\overline{g}^{-}_{i_{3}},\dots,\overline{g}^{-}_{i_{l}}$ nor a derivative of
$\overline{g}^{-}_{i_{2}}$ and $\overline{g}^{-}_{i_{2}}$ appears linearly in
it. Using the expression of $\overline{g}^{-}_{i_{1}}$ obtained before we can
write $\overline{g}^{-}_{i_{2}}$ as stated. Continuing in this way we get that
all $\overline{g}^{-}_{i_{1}},\dots,\overline{g}^{-}_{i_{l}}$ are contained in
$\mathcal{L}_{2}\langle\mathbf{\overline{t}}\rangle$.

Since $\mathbf{\overline{g}}^{0}$, $\mathbf{\overline{g}}^{-}$ are
differentially algebraically independent over $\mathcal{L}_{1}$ by Lemma 6.5
and we already proved that

$\mathcal{L}_{2}\langle\mathbf{\overline{t}}\rangle\,=\,\mathcal{L}_{1}\langle\mathbf{\overline{g}}^{0}\rangle\langle\overline{g}^{-}_{i}\in\mathbf{\overline{g}}^{-}\mid
i\notin\\{i_{1},\dots,i_{l}\\}\rangle\langle\mathbf{\overline{t}}\rangle\,=\,\mathcal{L}_{1}\langle\mathbf{\overline{g}}^{0},\mathbf{\overline{g}}^{-}\rangle\,,$

it follows from the invariance of the differential transcendence degree that
$\mathbf{\overline{t}}$ are differentially algebraically independent over
$\mathcal{L}_{2}$. ∎

In the proof of the following theorem we use Theorem 7.4 which will be
presented and proved in the next section.

######  Theorem 6.7.

There is a differential extension field $\mathcal{L}$ of
$C\langle\mathbf{a}^{+},\mathbf{a}^{-},\mathbf{a}^{0}\rangle$ with field of
constants $C$ such that the matrix
$A(\mathbf{a}^{+},\mathbf{a}^{-},\mathbf{a}^{0})$ is gauge equivalent to
$A_{G}(\mathbf{\overline{t}})$ over $\mathcal{L}$ for some
$\mathbf{\overline{t}}=(\overline{t}_{1},\dots,\overline{t}_{l})$ and the
differential Galois group of $A_{G}(\mathbf{\overline{t}})$ over $\mathcal{L}$
is the full group $G(C)$.

###### Proof.

We obtain from Lemma 6.4 that the matrix
$A(\mathbf{a}^{+},\mathbf{a}^{-},\mathbf{a}^{0})$ is gauge equivalent to
$A(\mathbf{\overline{h}}^{+},\mathbf{\overline{h}}^{-},\mathbf{\overline{h}}^{0})$
over $\mathcal{K}$ where
$\mathbf{\overline{h}}^{+}=(\overline{h}^{+}_{1},\dots,\overline{h}^{+}_{l},0,\dots,0)$
and the field of constants of $\mathcal{K}$ is $C$ by Lemma 6.3. Next Lemma
6.5 implies that there is an algebraic extension $\mathcal{L}$ of
$\mathcal{K}$ such that
$A(\mathbf{\overline{h}}^{+},\mathbf{\overline{h}}^{-},\mathbf{\overline{h}}^{0})$
is gauge equivalent to
$A(\mathbf{\overline{g}}^{+},\mathbf{\overline{g}}^{-},\mathbf{\overline{g}}^{0})$
over $\mathcal{L}$ where $\mathbf{\overline{g}}^{+}=(1,\dots,1,0,\dots,0)$.
Since $C$ is algebraically closed the field of constants of $\mathcal{L}$ is
also $C$. Finally Lemma 6.6 implies that
$A(\mathbf{\overline{g}}^{+},\mathbf{\overline{g}}^{-},\mathbf{\overline{g}}^{0})$
is gauge equivalent to the matrix $A_{G}(\mathbf{\overline{t}})$ over
$\mathcal{L}$ and that there is a subfield $\mathcal{L}_{2}$ of $\mathcal{L}$
such that $\mathcal{L}_{2}\langle\mathbf{\overline{t}}\rangle=\mathcal{L}$ and
the elements $\mathbf{\overline{t}}$ are differentially algebraically
independent over $\mathcal{L}_{2}$. Applying Theorem 7.4 to
$A_{G}(\mathbf{\overline{t}})$ and
$\mathcal{L}_{2}\langle\mathbf{\overline{t}}\rangle=\mathcal{L}$ we obtain
that the differential Galois group of $A_{G}(\mathbf{\overline{t}})$ over
$\mathcal{L}$ is $G(C)$. ∎

######  Remark 6.8.

Theorem 6.7 gives an alternative proof for the fact that the Galois group of
$A(\mathbf{a}^{+},\mathbf{a}^{-},\mathbf{a}^{0})$ over
$C\langle\mathbf{a}^{+},\mathbf{a}^{-},\mathbf{a}^{0}\rangle$ is $G(C)$ which
was already proven by L. Juan in [Jua04]. Indeed, since
$A(\mathbf{a}^{+},\mathbf{a}^{-},\mathbf{a}^{0})$ is gauge equivalent to
$A_{G}(\mathbf{\overline{t}})$ over $\mathcal{L}$, the Galois group of
$A(\mathbf{a}^{+},\mathbf{a}^{-},\mathbf{a}^{0})$ over $\mathcal{L}$ is also
$G(C)$. Because $G$ is an upper bound for the Galois group and
$A(\mathbf{a}^{+},\mathbf{a}^{-},\mathbf{a}^{0})$ has entries in
$C\langle\mathbf{a}^{+},\mathbf{a}^{-},\mathbf{a}^{0}\rangle$, the Galois
group of $A(\mathbf{a}^{+},\mathbf{a}^{-},\mathbf{a}^{0})$ over
$C\langle\mathbf{a}^{+},\mathbf{a}^{-},\mathbf{a}^{0}\rangle$ is $G(C)$ as
desired.

##  7\. The Galois Group of the Normal Form over
$F\langle\mathbf{\overline{t}}\rangle$

Let $F$ be an arbitrary differential field with field of constants $C$. For
differential indeterminates
$\mathbf{\overline{t}}=(\overline{t}_{1},\dots,\overline{t}_{l})$ over $F$ we
prove in this section that the Galois group of $A_{G}(\mathbf{\overline{t}})$
over $F\langle\mathbf{\overline{t}}\rangle$ is $G(C)$ by adapting to $F$ the
construction presented in [Seib] over $C$. To this end let
$\mathbf{v}=(v_{1},\dots,v_{l})$ be differential indeterminates over $F$. Then
$\mathbf{v}$ are differentially algebraically independent over $C$. The
results of [Seib] can be summarized in the following proposition.

######  Proposition 7.1.

Let $n(\overline{w})$ be a representative of the longest Weyl group element in
the normalizer of the torus. Then there are non-zero
$\mathbf{c}=(c_{1},\dots,c_{l})$ in $C$ and polynomials
$g_{1}(\mathbf{v}),\dots,g_{l}(\mathbf{v})\in C[\mathbf{v}]$ as well as
differential polynomials $f_{l+1}(\mathbf{v}),\dots,f_{m}(\mathbf{v})$ and
$s_{1}(\mathbf{v}),\dots,s_{l}(\mathbf{v})$ in $C\\{\mathbf{v}\\}$ having the
following properties:

  1. (a)

The polynomials $g_{1}(\mathbf{v}),\dots,g_{l}(\mathbf{v})$ are $C$-linearly
independent and homogeneous of degree one.

  2. (b)

Let $I_{\mathbf{s}}\subset C\\{\mathbf{v}\\}$ be the differential ideal
generated by $\mathbf{s}=(s_{1}(\mathbf{v}),\dots,s_{l}(\mathbf{v}))$. Then
the differential ring $C\\{\mathbf{v}\\}/I_{\mathbf{s}}$ is isomorphic to a
polynomial ring in finitely many variables which can be chosen to be the
images of $v_{1},\dots,v_{l}$ and finitely many of their derivatives. The
differential polynomials $\mathbf{s}$ are differentially algebraically
independent over $C$.

  3. (c)

The matrix

$A_{\rm
Liou}(\mathbf{v})\,=\,\sum_{i=1}^{l}g_{i}(\mathbf{v})H_{i}+\sum_{i=1}^{l}c_{i}X_{-\alpha_{i}}\in\mathfrak{b}^{-}(C\langle\mathbf{v}\rangle)$

defines a Picard-Vessiot extension $\mathcal{E}$ of
$C\langle\mathbf{v}\rangle$ with differential Galois group $B^{-}(C)$ and has
a fundamental matrix $Y_{\rm Liou}\in B^{-}(\mathcal{E})$.

  4. (d)

The logarithmic derivative of

$Y=\boldsymbol{u}\big{(}\mathbf{v},f_{l+1}(\mathbf{v}),\dots,f_{m}(\mathbf{v})\big{)}\,n(\overline{w})\,Y_{\rm
Liou}$

is $A_{G}(\mathbf{s})$ and $\mathcal{E}=C\langle\mathbf{v}\rangle(Y_{\rm
Liou})$ is a Picard-Vessiot extension of $C\langle\mathbf{s}\rangle$ for
$A_{G}(\mathbf{s})$ with differential Galois group $G(C)$.

###### Proof.

  1. (a)

The $g_{i}(\mathbf{v})$ correspond to the
$\overline{g}_{i}(\boldsymbol{\eta})$ defined on page 16 in [Seib] and the
desired properties follow from Lemma 5.3 (a) of [Seib].

  2. (b)

The $s_{i}(\mathbf{v})$ correspond to the $h_{j_{i}}(\boldsymbol{\eta})$
defined in Lemma 5.11 of [Seib] and so Theorem 5.13 of [Seib] yields that
$s_{1}(\mathbf{v}),\dots,s_{l}(\mathbf{v})$ are differentially algebraically
independent over $C$.

The properties of
$h_{j_{1}}(\boldsymbol{\eta}),\dots,h_{j_{l}}(\boldsymbol{\eta})$ stated in
[Seib, Lemma 5.11] imply that the differential ideal $I_{\mathbf{s}}$ is also
generated by $l$ differential polynomials
$\tilde{s}_{1}(\mathbf{v}),\dots,\tilde{s}_{l}(\mathbf{v})$ with the following
properties:

    1. (i)

Each $\tilde{s}_{i}(\mathbf{v})$ can be solved for some $v_{k_{i}}^{(d_{i})}$,
where $d_{i}\in\mathbb{Z}_{\geq 0}$.

    2. (ii)

The differential indeterminate $v_{k_{i}}$ appears in the remaining terms of
$\tilde{s}_{i}(\mathbf{v})$ with differentiation order lower than $d_{i}$.

    3. (iii)

The differential indeterminates $v_{k_{1}},\dots,v_{k_{l}}$ are pairwise
distinct.

We conclude that $C\\{\mathbf{v}\\}/I_{\mathbf{s}}$ is isomorphic to a
polynomial ring in finitely many variables, which we choose to be the images
of the indeterminates
$v_{k_{i}},v_{k_{i}}^{\prime},\dots,v_{k_{i}}^{(d_{i}-1)}$ with $1\leq i\leq
l$.

  3. (c)

This corresponds to Proposition 5.5 (a) and (b) in [Seib].

  4. (d)

This corresponds to Theorem 5.13 in [Seib]. ∎

######  Corollary 7.2.

We continue to use the notation of Proposition 7.1.

  1. (a)

Let $I_{\mathbf{s}}\subset F\\{\mathbf{v}\\}$ be the differential ideal
generated by $\mathbf{s}$. Then the differential ring
$F\\{\mathbf{v}\\}/I_{\mathbf{s}}$ is isomorphic to a polynomial ring in $m$
indeterminates which can be chosen to be the images of $v_{1},\dots,v_{l}$ and
finitely many of their derivatives.

  2. (b)

The differential polynomials $\mathbf{s}$ are differentially algebraically
independent over $F$.

###### Proof.

By Proposition 7.1 (b) the quotient $C\\{\mathbf{v}\\}/I_{\mathbf{s}}$ is
isomorphic to a polynomial ring $C[\mathbf{z}]$, where the indeterminates
$\mathbf{z}=(z_{1},\dots,z_{k})$ are the images of $\mathbf{v}$ and finitely
many of their derivatives, $k=d_{1}+d_{2}+\ldots+d_{l}$. Since $\mathbf{v}$
are differential indeterminates over $F$, the elements of

$\\{v_{i}^{(j)}\mid 1\leq i\leq l,\,j\in\mathbb{Z}_{\geq 0}\\}$

are algebraically independent over $F$. The same reasoning as in the proof of
Proposition 7.1 (b) shows that the images $\mathbf{z}$ in
$F\\{\mathbf{v}\\}/I_{\mathbf{s}}$ of the same
$v_{k_{i}},v_{k_{i}}^{\prime},\dots,v_{k_{i}}^{(d_{i}-1)}$ with $1\leq i\leq
l$ are algebraically independent over $F$. The algebraic independence of
$\mathbf{v}$ and $\mathbf{z}$ implies the isomorphisms

$F\\{\mathbf{v}\\}/I_{\mathbf{s}}\,\cong\,C\\{\mathbf{v}\\}/I_{\mathbf{s}}\otimes_{C}F\,\cong\,C[\mathbf{z}]\otimes_{C}F\,\cong\,F[\mathbf{z}].$

We prove that $k=m$. Again by Proposition 7.1 (b) the transcendence degree of
$C\langle\mathbf{v}\rangle$ over $C\langle\mathbf{s}\rangle$ is $k$. From
Proposition 7.1 (c) and (d) we obtain

$\mathrm{trdeg}_{C\langle\mathbf{v}\rangle}(\mathcal{E})=\dim(B^{-})=l+m$

and

$\mathrm{trdeg}_{C\langle\mathbf{s}\rangle}(\mathcal{E})=\dim(G)=l+2m.$

We conclude that
$\mathrm{trdeg}_{C\langle\mathbf{s}\rangle}(C\langle\mathbf{v}\rangle)=m$ and
so $k=m$. This proves (a).

Since the transcendence degree of $F\langle\mathbf{v}\rangle$ over
$F\langle\mathbf{s}\rangle$ is finite, the differential transcendence degree
of $F\langle\mathbf{v}\rangle$ over $F\langle\mathbf{s}\rangle$ is zero and so
the differential transcendence degree of $F\langle\mathbf{s}\rangle$ over $F$
is $l$ by [Kol73, II.9, Corollary 2]. As $F\langle\mathbf{s}\rangle$ is
generated by the $l$ elements $\mathbf{s}$ over $F$, they are differentially
algebraically independent over $F$. ∎

######  Proposition 7.3.

There is a Picard-Vessiot extension $\widehat{\mathcal{E}}$ of
$F\langle\mathbf{v}\rangle$ for $A_{\rm Liou}(\mathbf{v})$ with fundamental
matrix $\mathcal{Y}_{\rm Liou}\in B^{-}(\widehat{\mathcal{E}})$. The
differential Galois group of $\widehat{\mathcal{E}}$ over
$F\langle\mathbf{v}\rangle$ is $B^{-}(C)$.

###### Proof.

First we construct a Picard-Vessiot ring for $A_{\rm Liou}(\mathbf{v})$ in the
usual way. We extend the derivation of $F\langle\mathbf{v}\rangle$ to the ring

$F\langle\mathbf{v}\rangle[\mathrm{GL}_{n}]=F\langle\mathbf{v}\rangle[X,\det(X)^{-1}]$

by defining $X^{\prime}=A_{\rm Liou}(\mathbf{v})X$, where $X=(X_{ij})$ is a
matrix of $n^{2}$ indeterminates $X_{ij}$. Since $A_{\rm
Liou}(\mathbf{v})\in\mathfrak{b}^{-}(F\langle\mathbf{v}\rangle)$, the
embedding $(I)$ of the defining ideal $I\subset C[\mathrm{GL}_{n}]$ of the
Borel group $B^{-}$ is a differential ideal in
$F\langle\mathbf{v}\rangle[\mathrm{GL}_{n}]$. Let $J\subset
F\langle\mathbf{v}\rangle[\mathrm{GL}_{n}]$ be a maximal differential ideal
with $(I)\subseteq J$. Then the field of fractions $\widehat{\mathcal{E}}$ of
$F\langle\mathbf{v}\rangle[\mathrm{GL}_{n}]/J$ is a Picard-Vessiot extension
of $F\langle\mathbf{v}\rangle$ and the matrix of residue classes
$\mathcal{Y}_{\rm Liou}:=\overline{X}$ is a fundamental matrix contained in
$B^{-}(\widehat{\mathcal{E}})$.

It is left to show that the Galois group of $\widehat{\mathcal{E}}$ over
$F\langle\mathbf{v}\rangle$ is $B^{-}(C)$. To this end we apply [Kov69,
Theorem 2] to $A_{\rm Liou}(\mathbf{v})$. Clearly we have the semi-direct
product $B^{-}=T\,U^{-}$. The roots and root groups of the commutative
unipotent group $U^{-}/[U^{-},U^{-}]$ with respect to $T$ correspond to the
negative simple roots $-\alpha_{1},\dots,-\alpha_{l}$ and the root groups
$U_{-\alpha_{1}},\dots,U_{-\alpha_{l}}$ of $U^{-}$. Hence, the dimension of
each root group of $U^{-}/[U^{-},U^{-}]$ is one. Let $\chi_{1},\dots,\chi_{l}$
be the generators of the character group of $T$ and denote by
$e_{ij}\in\mathbb{Z}$ the exponent of $\chi_{j}$ when writing $-\alpha_{i}$ as
a product of $\chi_{1},\dots,\chi_{l}$. As in [Kov69, p. 596] we define the
map

$L_{g_{i}(\mathbf{v}),-\alpha_{i}}:F\langle\mathbf{v}\rangle\longrightarrow
F\langle\mathbf{v}\rangle,\quad x\longmapsto
x^{\prime}-\sum_{j=1}^{l}e_{ij}g_{j}(\mathbf{v})x,$

where $g_{1}(\mathbf{v})$, …, $g_{l}(\mathbf{v})$ have been defined in
Proposition 7.1, namely by

$A_{\rm
Liou}(\mathbf{v})\,=\,\sum_{i=1}^{l}g_{i}(\mathbf{v})H_{i}+\sum_{i=1}^{l}c_{i}X_{-\alpha_{i}}.$

Now Theorem 2 of [Kov69] implies that the differential Galois group of $A_{\rm
Liou}(\mathbf{v})$ is $B^{-}(C)$, if the images of
$g_{1}(\mathbf{v}),\dots,g_{l}(\mathbf{v})$ in the quotient
$F\langle\mathbf{v}\rangle/\ell\delta(F\langle\mathbf{v}\rangle)$ are
$\mathbb{Z}$-linearly independent, i.e. form a $\mathbb{Z}$-submodule of rank
$l$ and if $c_{i}$ does not reduce to zero in the quotient
$F\langle\mathbf{v}\rangle/L_{g_{i}(\mathbf{v}),-\alpha_{i}}(F\langle\mathbf{v}\rangle)$
for all $1\leq i\leq l$, i.e., the image of every $c_{i}$ generates a
$C$-vector space of dimension one. By Proposition 7.1 the elements
$\mathbf{c}$ are non-zero constants and
$g_{1}(\mathbf{v}),\dots,g_{l}(\mathbf{v})$ are $C$-linearly independent
homogeneous polynomials of degree one. We conclude that $\mathbf{c}$ and
$g_{1}(\mathbf{v}),\dots,g_{l}(\mathbf{v})$ satisfy the conditions of the
above mentioned theorem and so the differential Galois group of
$\widehat{\mathcal{E}}$ over $F\langle\mathbf{v}\rangle$ is $B^{-}(C)$. ∎

######  Theorem 7.4.

Let $F$ be a differential field with field of constants $C$ and let
$\mathbf{\overline{t}}=(t_{1},\dots,t_{l})$ be differential indeterminates
over $F$. Then the differential Galois group of $A_{G}(\mathbf{\overline{t}})$
over $F\langle\mathbf{\overline{t}}\rangle$ is $G(C)$.

###### Proof.

By Proposition 7.3 the matrix $A_{\rm Liou}(\mathbf{v})$ defines a Picard-
Vessiot extension $\widehat{\mathcal{E}}$ of $F\langle\mathbf{v}\rangle$ with
differential Galois group $B^{-}(C)$ and has a fundamental matrix
$\mathcal{Y}_{\rm Liou}\in B^{-}(\widehat{\mathcal{E}})$. As a first
consequence the transcendence degree of $\widehat{\mathcal{E}}$ over
$F\langle\mathbf{v}\rangle$ is

$\dim(B^{-})\,=\,m+l.$

Further, since the logarithmic derivatives of $\mathcal{Y}_{\rm Liou}$ and
$Y_{\rm Liou}$ are equal, Proposition 7.1 (d) implies that the logarithmic
derivative of

$\mathcal{Y}\,=\,\boldsymbol{u}\big{(}\mathbf{v},f_{l+1}(\mathbf{v}),\dots,f_{m}(\mathbf{v})\big{)}\,n(\overline{w})\,\mathcal{Y}_{\rm
Liou}$

is $A_{G}(\mathbf{s})$. By Corollary 7.2 (b) the differential polynomials
$\mathbf{s}$ are differentially algebraically independent over $F$. Moreover,
$\widehat{\mathcal{E}}$ is a Picard-Vessiot extension of
$F\langle\mathbf{s}\rangle$ for $A_{G}(\mathbf{s})$. Indeed, the fields of
constants of $\widehat{\mathcal{E}}$ and $F\langle\mathbf{s}\rangle$ are $C$,
since $\widehat{\mathcal{E}}$ is a Picard-Vessiot extension and
$F\langle\mathbf{s}\rangle$ is contained in $F\langle\mathbf{v}\rangle$. The
matrix $\mathcal{Y}$ is a fundamental matrix for $A_{G}(\mathbf{s})$ and its
entries clearly lie in $\widehat{\mathcal{E}}$. In order to prove that
$\mathcal{Y}_{ij}$ generate
$\widehat{\mathcal{E}}=F\langle\mathbf{v}\rangle(\mathcal{Y}_{\rm Liou})$ as a
field over $F\langle\mathbf{s}\rangle$, we show the non-trivial inclusion
$\widehat{\mathcal{E}}\subseteq F\langle\mathbf{s}\rangle(\mathcal{Y}_{ij})$.
Since $\mathcal{Y}\in G(\widehat{\mathcal{E}})$, the Bruhat decomposition and
its uniqueness imply that we can express $\mathbf{v}$,
$f_{l+1}(\mathbf{v}),\dots,f_{m}(\mathbf{v})$ and the entries of
$\mathcal{Y}_{\rm Liou}$ as rational functions in $\mathcal{Y}_{ij}$. Since
$F\langle\mathbf{s}\rangle(\mathcal{Y}_{ij})$ is a differential field, the
derivatives of $\mathcal{Y}_{ij}$ being defined by $A_{G}(\mathbf{s})$, it
contains besides the entries of $\mathcal{Y}_{\rm Liou}$ all derivatives of
$\mathbf{v}$ and so $\widehat{\mathcal{E}}\subseteq
F\langle\mathbf{s}\rangle(\mathcal{Y}_{ij})$. Hence,
$\widehat{\mathcal{E}}=F\langle\mathbf{s}\rangle(\mathcal{Y}_{ij})$ and
$\widehat{\mathcal{E}}$ is a Picard-Vessiot extension of
$F\langle\mathbf{s}\rangle$ for $A_{G}(\mathbf{s})$.

$\widehat{\mathcal{E}}$$F\langle\mathbf{v}\rangle$$F\langle\mathbf{s}\rangle$${\rm
trdeg}=m+l$${\rm trdeg}=m$

By Corollary 7.2 (a) the transcendence degree of $F\langle\mathbf{v}\rangle$
over $F\langle\mathbf{s}\rangle$ is $m$ and so the transcendence degree of
$\widehat{\mathcal{E}}$ over $F\langle\mathbf{s}\rangle$ is

$2m+l\,=\,\mathrm{dim}(G).$

We conclude that the differential Galois group of $\widehat{\mathcal{E}}$ over
$F\langle\mathbf{s}\rangle$ for $A_{G}(\mathbf{s})$ is $G(C)$. Since
$\mathbf{s}$ are differentially algebraically independent over $F$, the Galois
group of $A_{G}(\mathbf{\overline{t}})$ over
$F\langle\mathbf{\overline{t}}\rangle$ is $G(C)$, where
$\mathbf{\overline{t}}$ are differential indeterminates over $F$. ∎

##  8\. Specializing the Coefficients of the Generic Equation

Let $\mathcal{L}$ be the extension field of
$C\langle\mathbf{a}^{+},\mathbf{a}^{-},\mathbf{a}^{0}\rangle$ constructed in
the previous sections with respect to the resolving Weyl group element
$\overline{w}$, i.e. the longest Weyl group element. Note that in this case we
have

$\Psi\,=\,\\{-\alpha_{l+1},\dots,-\alpha_{m}\\}\,,\quad\mathbf{b}_{\overline{w}}\,=\,(b_{l+1},\dots,b_{m})\,,$

and that $\mathcal{L}$ is an algebraic extension of the differential field

$\mathcal{K}\,=\,\operatorname{Frac}(\mathcal{R})\,=\,C\langle\mathbf{\overline{b}}_{\overline{w}},\mathbf{\overline{a}}^{+},\mathbf{\overline{a}}^{-},\mathbf{\overline{a}}^{0}\rangle,$

where we denote by $\mathcal{R}$ the differential ring

(11)
$\mathcal{R}\,=\,C\\{\mathbf{b}_{\overline{w}},\mathbf{a}^{+},\mathbf{a}^{-},\mathbf{a}^{0}\\}/I_{\overline{w}}\,=\,C\\{\mathbf{\overline{b}}_{\overline{w}},\mathbf{\overline{a}}^{+},\mathbf{\overline{a}}^{-},\mathbf{\overline{a}}^{0}\\}.$

Let $\mathcal{M}$ be the multiplicatively closed subset generated by
$\overline{h}^{+}_{1},\dots,\overline{h}^{+}_{l}$ and let
$\boldsymbol{\overline{x}}=(\overline{x}_{1},\dots,\overline{x}_{l})$ be the
solutions of (10) introduced in the proof of Lemma 6.5, i.e., each
$\overline{x}_{i}$ is a certain product of powers of
$\overline{h}^{+}_{1},\dots,\overline{h}^{+}_{l}$ with rational exponents.
Then

$\mathcal{S}\,:=\,(\mathcal{M}^{-1}\mathcal{R})[\boldsymbol{\overline{x}}]$

defines an integral ring extension $\mathcal{S}$ of
$\mathcal{M}^{-1}\mathcal{R}$ and the gauge transformation of
$A(\mathbf{\overline{a}}^{+},\mathbf{\overline{a}}^{-},\mathbf{\overline{a}}^{0})$
to $A_{G}(\mathbf{\overline{t}})$ as in Theorem 6.7 is defined over
$\mathcal{S}$. Indeed, following the proof of the theorem, the matrix
$A(\mathbf{\overline{a}}^{+},\mathbf{\overline{a}}^{-},\mathbf{\overline{a}}^{0})$
is gauge equivalent to
$A(\mathbf{\overline{h}}^{+},\mathbf{\overline{h}}^{-},\mathbf{\overline{h}}^{0})$
over $\mathcal{R}$, where
$\mathbf{\overline{h}}^{+}=(\overline{h}^{+}_{1},\dots,\overline{h}^{+}_{l},0,\dots,0)\in\mathcal{R}^{m}$
with non-zero $\overline{h}^{+}_{1},\dots,\overline{h}^{+}_{l}$. Since
$\mathcal{S}$ contains the roots for the normalization of the coefficients
$\overline{h}^{+}_{1},\dots,\overline{h}^{+}_{l}$, it follows that
$A(\mathbf{\overline{h}}^{+},\mathbf{\overline{h}}^{-},\mathbf{\overline{h}}^{0})$
is gauge equivalent to
$A(\mathbf{\overline{g}}^{+},\mathbf{\overline{g}}^{-},\mathbf{\overline{g}}^{0})$
with $\mathbf{\overline{g}}^{+}=(1,\dots,1,0,\dots,0)$. The transformation
from
$A(\mathbf{\overline{g}}^{+},\mathbf{\overline{g}}^{-},\mathbf{\overline{g}}^{0})$
to $A_{G}(\mathbf{\overline{t}})$ needs no further inverse elements nor
algebraic extensions and is therefore also defined over $\mathcal{S}$.

Let $F$ be an arbitrary differential field with field of constants $C$ and let

$A\,=\,A(\mathbf{a}^{+}_{*},\mathbf{a}^{-}_{*},\mathbf{a}^{0}_{*})\in\mathfrak{g}(F)$

define a Picard-Vessiot extension $E$ with Galois group $G(C)$. For
differential indeterminates $\mathbf{b}_{*}=((b_{*})_{l+1},\dots,(b_{*})_{m})$
over $E$ we denote by $\sigma$ the homomorphism of differential rings

$\sigma:C\\{\mathbf{b}_{\overline{w}},\mathbf{a}^{+},\mathbf{a}^{-},\mathbf{a}^{0}\\}\longrightarrow
E\\{\mathbf{b}_{*}\\},\quad(\mathbf{b}_{\overline{w}},\mathbf{a}^{+},\mathbf{a}^{-},\mathbf{a}^{0})\longmapsto(\mathbf{b}_{*},\mathbf{a}^{+}_{*},\mathbf{a}^{-}_{*},\mathbf{a}^{0}_{*}).$

We fix this notation for the rest of the section.

######  Theorem 8.1.

Suppose $A=A(\mathbf{a}^{+}_{*},\mathbf{a}^{-}_{*},\mathbf{a}^{0}_{*})$
defines a Picard-Vessiot extension $E$ of $F$ with differential Galois group
$G(C)$. If $\sigma(I_{\overline{w}})$ is a maximal differential ideal in
$E\\{\mathbf{b}_{*}\\}$, then there exists a differential extension field $L$
of $F$ with the following properties:

  1. (a)

The field of constants of $L$ is $C$.

  2. (b)

There exists a differential homomorphism
$\overline{\sigma}:\mathcal{S}\rightarrow L$ such that

$\overline{\sigma}(A(\mathbf{\overline{a}}^{+},\mathbf{\overline{a}}^{-},\mathbf{\overline{a}}^{0}))\,=\,A.$

  3. (c)

The matrix $A$ is gauge equivalent over $L$ to the specialization
$\overline{\sigma}(A_{G}(\mathbf{\overline{t}}))$ of
$A_{G}(\mathbf{\overline{t}})$.

  4. (d)

The differential Galois group of $A$ over $L$ is $G(C)$.

###### Proof.

First we construct a purely transcendental differential field extension
$F\langle\mathbf{\overline{b}_{*}}\rangle$ of $F$ with constants $C$ and show
that the Galois group over $F\langle\mathbf{\overline{b}_{*}}\rangle$ is still
the full group $G$. Since $\sigma(I_{\overline{w}})$ is a maximal differential
ideal in $E\\{\mathbf{b}_{*}\\}$, the field of constants of

$E\langle\mathbf{\overline{b}_{*}}\rangle\,=\,\operatorname{Frac}(E\\{\mathbf{b}_{*}\\}/\sigma(I_{\overline{w}}))$

is $C$. Thus the fields of constants of the subring

$E\\{\mathbf{\overline{b}_{*}}\\}\,=\,E\\{\mathbf{b}_{*}\\}/\sigma(I_{\overline{w}})$

and of the subfield $F\langle\mathbf{\overline{b}_{*}}\rangle$ are also $C$.
Hence, the differential field extension
$E\langle\mathbf{\overline{b}_{*}}\rangle$ of
$F\langle\mathbf{\overline{b}_{*}}\rangle$ is a Picard-Vessiot extension for
$A$. Since $A\in\mathfrak{g}(F)$, its differential Galois group $H(C)$ is a
subgroup of $G(C)$. We prove that $H(C)=G(C)$. Recall from Section 5 that
$I_{\overline{w}}$ is generated by
$f^{+}_{\overline{w},l+1},\dots,f^{+}_{\overline{w},m}$ and by Theorem 5.5 the
leader of $f^{+}_{\overline{w},i}$ is $b_{i}^{\prime}$ and
$f^{+}_{\overline{w},i}$ is linear in its leader and has constant initial.
Then the same is true for the generators
$\sigma(f^{+}_{\overline{w},l+1}),\dots,\sigma(f^{+}_{\overline{w},m})$ of
$\sigma(I_{\overline{w}})$ and, since $\sigma(I_{\overline{w}})$ is a maximal
differential ideal, the transcendence degree of
$E\langle\mathbf{\overline{b}_{*}}\rangle$ over $E$ is $m-l$. Moreover, the
assumption on the Galois group implies

$\mathrm{trdeg}_{F}(E)\,=\,\dim(G)\,=\,2m+l$

and so the transcendence degree of $E\langle\mathbf{\overline{b}_{*}}\rangle$
over $F$ is $3m$. Again by the properties of
$\sigma(f^{+}_{\overline{w},l+1}),\dots,\sigma(f^{+}_{\overline{w},m})$ and
the maximality of $\sigma(I_{\overline{w}})$ the transcendence degree of
$F\langle\mathbf{\overline{b}_{*}}\rangle$ over $F$ is also $m-l$. It follows
that

$\dim(H(C))\,=\,\mathrm{trdeg}_{F\langle\mathbf{\overline{b}_{*}}\rangle}(E\langle\mathbf{\overline{b}_{*}}\rangle)\,=\,2m+l\,=\,\dim(G(C))$

and thus, by the main theorem of Galois theory, we conclude that $H(C)=G(C)$.

$E\langle\mathbf{\overline{b}_{*}}\rangle$$E$$F$${\rm trdeg}=m-l$${\rm
trdeg}=2m+l$$E\langle\mathbf{\overline{b}_{*}}\rangle$$F\langle\mathbf{\overline{b}_{*}}\rangle$$F$${\rm
trdeg}=2m+l$${\rm trdeg}=m-l$

Clearly the differential homomorphism

$\widetilde{\sigma}:\mathcal{R}\longrightarrow
F\langle\mathbf{\overline{b}_{*}}\rangle,\quad(\mathbf{\overline{a}}^{+},\mathbf{\overline{a}}^{-},\mathbf{\overline{a}}^{0},\mathbf{\overline{b}}_{\overline{w}})\longmapsto(\mathbf{a}^{+}_{*},\mathbf{a}^{-}_{*},\mathbf{a}^{0}_{*},\mathbf{\overline{b}_{*}}),$

where $\mathcal{R}$ is defined in (11), is well-defined and satisfies
$\widetilde{\sigma}(A(\mathbf{\overline{a}}^{-},\mathbf{\overline{a}}^{0},\mathbf{\overline{a}}^{+}))=A$.
Because
$A(\mathbf{\overline{a}}^{-},\mathbf{\overline{a}}^{0},\mathbf{\overline{a}}^{+})$
can be gauge transformed to
$A(\mathbf{\overline{h}}^{+},\mathbf{\overline{h}}^{-},\mathbf{\overline{h}}^{0})$
over $\mathcal{R}$ and $\widetilde{\sigma}$ is a differential homomorphism,
$A$ is gauge equivalent to
$\widetilde{\sigma}(A(\mathbf{\overline{h}}^{+},\mathbf{\overline{h}}^{-},\mathbf{\overline{h}}^{0}))$
over $F\langle\mathbf{\overline{b}_{*}}\rangle$. The fact that the
differential Galois group of $E\langle\mathbf{\overline{b}_{*}}\rangle$ over
$F\langle\mathbf{\overline{b}_{*}}\rangle$ is $G(C)$ implies that
$\widetilde{\sigma}(\overline{h}^{+}_{1}),\dots,\widetilde{\sigma}(\overline{h}^{+}_{l})$
are all non-zero, since otherwise we would have obtained a gauge equivalent
matrix in a proper Lie subalgebra of $\mathfrak{g}$. The extension of
$\widetilde{\sigma}$ to
$\mathcal{S}=(\mathcal{M}^{-1}\mathcal{R})[\boldsymbol{\overline{x}}]$ depends
on the existence of appropriate images of $\boldsymbol{\overline{x}}$. Since
$\widetilde{\sigma}(\overline{h}^{+}_{1}),\dots,\widetilde{\sigma}(\overline{h}^{+}_{l})$
are non-zero, the specialization of the equations in (10) by
$\widetilde{\sigma}$ has a solution in an algebraic extension field. Hence,
there exists an algebraic extension $L$ of
$F\langle\mathbf{\overline{b}_{*}}\rangle$ and a differential homomorphism

$\overline{\sigma}:\mathcal{S}\longrightarrow L$

extending $\widetilde{\sigma}$. The gauge equivalence of
$A(\mathbf{\overline{h}}^{+},\mathbf{\overline{h}}^{-},\mathbf{\overline{h}}^{0})$
to $A_{G}(\mathbf{\overline{t}})$ over $\mathcal{S}$ implies the gauge
equivalence of $A$ to $\overline{\sigma}(A_{G}(\mathbf{\overline{t}}))$ over
$L$. Finally, since $G(C)$ is a connected group and $L$ is an algebraic
extension of $F\langle\mathbf{\overline{b}_{*}}\rangle$, the differential
Galois group for $A$ remains $G(C)$ over $L$. ∎

Finally, for a Picard-Vessiot extension $E$ of $F$ with differential Galois
group $\mathrm{SL}_{3}(C)$ and defining matrix

$A\,=\,\sum_{i=1}^{3}(a_{*})^{-}_{i}X_{-\alpha_{i}}+\sum_{i=1}^{2}(a_{*})^{0}_{i}H_{i}+\sum_{i=1}^{3}(a_{*})^{+}_{i}X_{\alpha_{i}}$

we show how Theorem 8.1 allows to construct a differential extension field $L$
of $F$ such that the field of constants of $L$ is $C$, $A$ is gauge equivalent
to a matrix in normal form over $L$ and the differential Galois group of $A$
over $L$ is the full group $\mathrm{SL}_{3}(C)$. To this end let
$\boldsymbol{r}=(r_{1},r_{2})$ be two differential indeterminates over $E$.
Then clearly $E\langle\boldsymbol{r}\rangle$ is a Picard-Vessiot extension of
$F\langle\boldsymbol{r}\rangle$ with differential Galois group
$\mathrm{SL}_{3}(C)$. Gauge transformation of $A$ by
$u_{-\alpha_{1}}(r_{1})u_{-\alpha_{2}}(r_{2})$ yields a new defining matrix

$A_{\boldsymbol{r}}=A(\boldsymbol{a}^{+}_{\boldsymbol{r}},\boldsymbol{a}^{-}_{\boldsymbol{r}},\boldsymbol{a}^{0}_{\boldsymbol{r}})\in\mathfrak{sl}_{3}(F\langle\boldsymbol{r}\rangle)$

for the extension $E\langle\boldsymbol{r}\rangle$ of
$F\langle\boldsymbol{r}\rangle$. As in the paragraph before Theorem 8.1 with
$A$, $F$ and $E$ replaced by $A_{\boldsymbol{r}}$,
$F\langle\boldsymbol{r}\rangle$ and $E\langle\boldsymbol{r}\rangle$,
respectively, we consider the differential ring homomorphism

$\sigma:C\\{\mathbf{b}_{\overline{w}},\mathbf{a}^{+},\mathbf{a}^{-},\mathbf{a}^{0}\\}\longrightarrow
E\langle\boldsymbol{r}\rangle\\{\mathbf{b}_{*}\\},\quad(\mathbf{b}_{\overline{w}},\mathbf{a}^{+},\mathbf{a}^{-},\mathbf{a}^{0})\longmapsto(\mathbf{b}_{*},\boldsymbol{a}^{+}_{\boldsymbol{r}},\boldsymbol{a}^{-}_{\boldsymbol{r}},\boldsymbol{a}^{0}_{\boldsymbol{r}}).$

Once we have verified for $A_{\boldsymbol{r}}$ and
$E\langle\boldsymbol{r}\rangle/F\langle\boldsymbol{r}\rangle$ the condition on
the differential ideal $\sigma(I_{\overline{w}})$, Theorem 8.1 yields the
desired extension $L$ of $F\langle\boldsymbol{r}\rangle$ and hence of $F$.

In order to identify the ideal $\sigma(I_{\overline{w}})$, we note that for
$\mathrm{SL}_{3}$ and the longest Weyl group element $\overline{w}$ we have
$\Psi=\\{-\alpha_{3}\\}$ and $\mathbf{b}_{\overline{w}}=(b_{3})$. Gauge
transformation with $u_{-\alpha_{3}}((b_{*})_{3})$ yields then the system

$\sigma(S_{\overline{w}}):\quad\sigma(f^{+}_{\overline{w},3})=0$

consisting of a single equation, where $\sigma(f^{+}_{\overline{w},3})$ is

$\displaystyle(b_{*})_{3}^{\prime}-(a_{*})^{+}_{3}(b_{*})_{3}^{2}+((a_{*})^{+}_{3}r_{1}r_{2}-r_{2}(a_{*})^{+}_{2}-r_{1}(a_{*})^{+}_{1}+(a_{*})^{0}_{1}+(a_{*})^{0}_{2})(b_{*})_{3}\,+$
$\displaystyle
r_{2}(a_{*})^{-}_{1}+(a_{*})^{-}_{3}-(r_{2}((a_{*})^{0}_{2}-(a_{*})^{0}_{1})+(a_{*})^{-}_{2})r_{1}+(r_{2}(a_{*})^{+}_{2}-(a_{*})^{0}_{2})r_{1}r_{2}-r_{1}r_{2}^{\prime}.$

The next lemma gives a criterion for the ideal
$\sigma(I_{\overline{w}})=\langle\sigma(f^{+}_{\overline{w},3})\rangle$ to be
a maximal differential ideal in
$E\langle\boldsymbol{r}\rangle\\{(b_{*})_{3}\\}$.

######  Lemma 8.2.

Let $K$ be a differential field of characteristic zero. Let

$f(y)=y^{\prime}+c_{3}y^{3}+c_{2}y^{2}+c_{1}y+c_{0}\in K\\{y\\},\quad
c_{0}\neq 0,$

be a differential polynomial. For every positive integer $m$ we define the
system $\Sigma_{m}$ of $m$ differential equations in $m$ differential
indeterminates $y_{0}$, …, $y_{m-1}$

$\left\\{\begin{array}[]{rcl}y_{0}^{\prime}\\!\\!&\\!\\!=\\!\\!&\\!\\!c_{3}((2y_{m-2}-y_{m-1}^{2})y_{0}-my_{m-3})+(c_{2}y_{m-1}-mc_{1})y_{0}+c_{0}y_{1},\\\
\\!\\!&\\!\\!\vdots\\!\\!&\\!\\!\\\
y_{k}^{\prime}\\!&\\!=\\!&\\!c_{3}((2y_{m-2}-y_{m-1}^{2})y_{k}-my_{m-3}+ky_{k-2}+y_{m-1}y_{k-1}-2y_{k-2})\\\\[1.99997pt]
\\!\\!&\\!\\!\\!\\!&\\!\\!+\,(c_{2}y_{m-1}-mc_{1}+kc_{1})y_{k}+(k-m-1)c_{2}y_{k-1}+(k+1)c_{0}y_{k+1},\\\
\\!\\!&\\!\\!\vdots\\!\\!&\\!\\!\\\
y_{m-1}^{\prime}\\!\\!&\\!\\!=\\!\\!&\\!\\!c_{3}((3y_{m-2}-y_{m-1}^{2})y_{m-1}-3y_{m-3})\\\\[1.99997pt]
\\!\\!&\\!\\!\\!\\!&\\!\\!+\,(c_{2}y_{m-1}-c_{1})y_{m-1}-2c_{2}y_{m-2}+mc_{0},\end{array}\right.$

where the convention $y_{m}=1$, $y_{-2}=0$, $y_{-1}=0$ is used. If for all
positive integers $m$ the system has no solution in $K^{m}$, then the
differential ideal $\langle f(y)\rangle$ is a maximal differential ideal in
$K\\{y\\}$.

###### Proof.

Let $g(y)\in K\\{y\\}$ be a differential polynomial such that

$\langle f(y)\rangle\subsetneq\langle f(y),g(y)\rangle\subseteq K\\{y\\}.$

We can assume without loss of generality that $g(y)$ is a polynomial in
$K[y]$. Indeed, from $f(y)$ and all its derivatives one can express all
$y^{(n)}$ in terms of polynomials in $K[y]$ modulo $\langle f(y)\rangle$.
Hence, we can eliminate all derivatives in $g(y)$ and obtain a non-zero
polynomial in $K[y]$, since otherwise we would have $\langle
f(y)\rangle=\langle f(y),g(y)\rangle$. So let

$g(y)\,=\,\sum_{k=0}^{m}a_{k}y^{k}\quad\mathrm{with}\quad a_{m}=1.$

We prove by induction on the degree $m\geq 1$ that $\langle
f(y),g(y)\rangle=K\\{y\\}$.

Let $m=1$, i.e. $g(y)=y+a_{0}$. Differentiating $g(y)$ and subtracting $f(y)$
yields

$\tilde{g}(y)\,=\,y^{\prime}+a_{0}^{\prime}-f(y)\,=\,a_{0}^{\prime}-(c_{3}y^{3}+c_{2}y^{2}+c_{1}y+c_{0}).$

In the next step we add $c_{3}y^{2}g(y)$ to $\tilde{g}(y)$ and then subtract
$(c_{3}a_{0}-c_{2})yg(y)$ and $(-c_{3}a_{0}^{2}+c_{2}a_{0}-c_{1})g(y)$ from
the result. This leads to

$g_{0}\,:=\,a_{0}^{\prime}+c_{3}a_{0}^{3}-c_{2}a_{0}^{2}+c_{1}a_{0}-c_{0}\in\langle
f(y),g(y)\rangle.$

Since by assumption the equation

$y_{0}^{\prime}\,=\,-c_{3}y_{0}^{3}+c_{2}y_{0}^{2}-c_{1}y_{0}+c_{0}$

has no solution in $K$, we obtain $g_{0}\in K\setminus\\{0\\}$ and so $\langle
f(y),g(y)\rangle=K\\{y\\}$.

Let $m\geq 2$. Differentiating $g(y)$ yields

$\displaystyle
g(y)^{\prime}\,=\,my^{m-1}y^{\prime}+\big{(}\sum_{k=1}^{m-1}ka_{k}y^{k-1}y^{\prime}+a_{k}^{\prime}y^{k}\big{)}+a_{0}^{\prime}.$

We are going to reduce $g(y)^{\prime}$ with differential polynomials in
$\langle f(y),g(y)\rangle$ until we obtain a polynomial of degree $m-1$ which
we prove to be non-zero. In order to eliminate the derivative $y^{\prime}$ in
$g(y)^{\prime}$ we subtract $my^{m-1}f(y)$ and

$\sum_{k=1}^{m-1}ky^{k-1}a_{k}f$

from $g(y)^{\prime}$ and obtain a polynomial in $\langle f(y),g(y)\rangle$ of
degree $m+2$. More precisely,

$\displaystyle\tilde{g}_{1}(y)\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!-mc_{3}y^{m+2}-(mc_{2}+(m-1)a_{m-1}c_{3})y^{m+1}-$
$\displaystyle\\!(mc_{1}+(m-1)a_{m-1}c_{2}+(m-2)a_{m-2}c_{3})y^{m}+$
$\displaystyle\\!\sum_{k=0}^{m-1}\big{(}a_{k}^{\prime}-(k-2)a_{k-2}c_{3}-(k-1)a_{k-1}c_{2}-ka_{k}c_{1}-(k+1)a_{k+1}c_{0}\big{)}y^{k}$

with the convention $a_{-2}=a_{-1}=0$. In the next step we eliminate the term
$-mc_{3}y^{m+2}$ in $\tilde{g}_{1}(y)$. To this end we subtract
$-mc_{3}y^{2}g(y)$ from $\tilde{g}_{1}(y)$. This yields

$\displaystyle\tilde{g}_{2}(y)\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!(-mc_{2}+a_{m-1}c_{3})y^{m+1}+(-mc_{1}-(m-1)a_{m-1}c_{2}+2a_{m-2}c_{3})y^{m}+$
$\displaystyle\\!\sum_{k=0}^{m-1}[a_{k}^{\prime}+c_{3}a_{m-3}m+c_{0}(-k-1)a_{k+1}-c_{1}ka_{k}-c_{2}a_{k-1}(k-1)$
$\displaystyle\\!\qquad+\,c_{3}a_{k-2}(2-k)]y^{k}.$

In a further elimination step we get rid of the term of degree $m+1$. We
subtract from $\tilde{g}_{2}(y)$ the expression
$(-mc_{2}-(m-1)a_{m-1}c_{3}+mc_{3}a_{m-1})yg(y)$. This leads to

$\displaystyle\tilde{g}_{3}(y)\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!(-c_{1}m-c_{3}a_{m-1}^{2}+c_{2}a_{m-1}+2c_{3}a_{m-2})y^{m}+$
$\displaystyle\\!\sum_{k=0}^{m-1}\big{[}a_{k}^{\prime}+(c_{3}a_{m-3}+c_{2}a_{k-1})m-c_{3}a_{k-1}a_{m-1}-c_{0}(k+1)a_{k+1}$
$\displaystyle\\!\qquad-\,c_{1}ka_{k}+(-c_{2}a_{k-1}-c_{3}a_{k-2})k+c_{2}a_{k-1}+2c_{3}a_{k-2}\big{]}y^{k}.$

Finally, we subtract from $\tilde{g}_{3}(y)$ the expression

$\displaystyle(-c_{1}m-c_{3}a_{m-1}^{2}+c_{2}a_{m-1}+2c_{3}a_{m-2})g(y)$

and obtain the polynomial

$\displaystyle\tilde{g}_{4}(y)\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\sum_{k=0}^{m-1}\big{[}a_{k}^{\prime}+((a_{m-1}^{2}-2a_{m-2})a_{k}-ka_{k-2}-a_{m-1}a_{k-1}+a_{m-3}m+2a_{k-2})c_{3}$
$\displaystyle\\!\qquad-\,(a_{m-1}c_{2}+c_{1}(k-m))a_{k}-c_{2}(k-m-1)a_{k-1}-c_{0}(k+1)a_{k+1}\big{]}y^{k}$
$\displaystyle\\!=\\!$
$\displaystyle\\!\sum_{k=0}^{m-1}\big{[}a_{k}^{\prime}-c_{3}((2a_{m-2}-a_{m-1}^{2})a_{k}-ma_{m-3}+ka_{k-2}+a_{m-1}a_{k-1}-2a_{k-2})$
$\displaystyle\\!\qquad-\,(c_{2}a_{m-1}-mc_{1}+c_{1}k)a_{k}-(k-m-1)c_{2}a_{k-1}-(k+1)c_{0}a_{k+1}\big{]}y^{k},$

which is of degree at most $m-1$. In order for $\tilde{g}_{4}(y)$ to be zero,
all coefficients have to be zero. But this corresponds to a solution

$(\overline{y}_{0},\dots,\overline{y}_{m-1})\,=\,(a_{0},\dots a_{m-1})\in
K^{m}$

of the system of differential equations in the statement of the lemma. Since
by assumption such a solution does not exists, $\tilde{g}_{4}(y)$ is not zero
and so by the induction hypothesis, $\langle f(y),g(y)\rangle=\langle
f(y),\tilde{g}_{4}(y)\rangle=K\\{y\\}$. ∎

According to the last lemma the ideal
$\sigma(I_{\overline{w}})=\langle\sigma(f^{+}_{\overline{w},3})\rangle$ is a
maximal differential ideal in
$E\langle\boldsymbol{r}\rangle\\{(b_{*})_{3}\\}$, if for all $m>0$ the system
of differential equations in Lemma 8.2 over $E\langle\boldsymbol{r}\rangle$
has no solutions in $E\langle\boldsymbol{r}\rangle$. The remainder of this
section is dedicated to the verification of this condition.

######  Lemma 8.3.

Let $K$ be a differential field of characteristic zero and let $r$ be a
differential indeterminate over $K$. Let $a$, $b\in K\\{r\\}$, $b\neq 0$, such
that

$\frac{a}{b}\notin K(r^{(j)}\mid 0\leq j\leq d-1),$

where $d=\mathrm{max}\\{\mathrm{ord}(a),\mathrm{ord}(b)\\}$. Then
$\mathrm{ord}(a^{\prime}b-ab^{\prime})=d+1$.

###### Proof.

First assume $\mathrm{ord}(a)>\mathrm{ord}(b)$. Then the derivative of $a$ is

$a^{\prime}=\mathrm{sep}(a)r^{(d+1)}+g_{a},\qquad\mathrm{sep}(a)\,=\,\partial_{r^{(d)}}(a),$

where $g_{a}\in K[r^{(j)}\mid 0\leq j\leq d]$ and
$\partial_{r^{(d)}}=\partial/\partial r^{(d)}$. Since $a\notin K$, we have
$\mathrm{sep}(a)\neq 0$. By assumption $\mathrm{ord}(b)<d$ and so we have
$\mathrm{ord}(ab^{\prime})\leq d$. With $\mathrm{ord}(a^{\prime})=d+1$ we
conclude that $\mathrm{ord}(a^{\prime}b)=d+1$ and so
$\mathrm{ord}(a^{\prime}b-b^{\prime}a)=d+1$.

In case $\mathrm{ord}(a)<\mathrm{ord}(b)$ the claim follows as in the previous
case.

Finally assume $\mathrm{ord}(a)=\mathrm{ord}(b)$. We compute

$a^{\prime}b-ab^{\prime}\,=\,r^{(d+1)}(\mathrm{sep}(a)b-\mathrm{sep}(b)a)+g$

with $g\in K[r^{(j)}\mid 0\leq j\leq d]$. We consider now the expression
$\mathrm{sep}(a)b-\mathrm{sep}(b)a$ as an element of the differential ring

$K[r^{(j)}\mid 0\leq j\leq d-1][r^{(d)}]$

with derivation $\partial_{r^{(d)}}$. Recognizing the coefficient of
$r^{(d+1)}$ as a Wronskian determinate, we obtain that

$\mathrm{sep}(a)b-\mathrm{sep}(b)a\,=\,0$

if and only if $a,b$ are $K(r^{(j)}\mid 0\leq j\leq d-1)$-linearly dependent,
i.e. $a=eb$ with $e\in K(r^{(j)}\mid 0\leq j\leq d-1])$, which is not the case
by assumption. ∎

######  Lemma 8.4.

Let $a$, $b\in E[r_{1},r_{2}]\subset E\\{r_{1},r_{2}\\}$. Then in
$a^{\prime}b-ab^{\prime}$ the indeterminate $r_{1}^{\prime}$ does not appear
if and only if $a/b\in E(r_{2})$.

###### Proof.

The derivatives of $a$ and $b$ are

$\displaystyle a^{\prime}\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!r_{1}^{\prime}\partial_{r_{1}}(a)+g_{a},$ $\displaystyle
b^{\prime}\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!r_{1}^{\prime}\partial_{r_{1}}(b)+g_{b},$

where $g_{a}$, $g_{b}\in E[r_{1},r_{2},r_{2}^{\prime}]$ and
$\partial_{r_{1}}=\partial/\partial r_{1}$. This yields

$a^{\prime}b-ab^{\prime}\,=\,r_{1}^{\prime}\big{(}\partial_{r_{1}}(a)b-a\partial_{r_{1}}(b)\big{)}+(g_{a}b-ag_{b}).$

We consider $a,b$ as elements of the differential ring $E[r_{2}][r_{1}]$ with
derivation $\partial_{r_{1}}$. Then, recognizing the coefficient of
$r_{1}^{\prime}$ as a Wronskian determinant, we have

$\partial_{r_{1}}(a)b-a\partial_{r_{1}}(b)\,=\,0$

if and only if $a,b$ are $E(r_{2})$-linearly dependent, i.e. $a=eb$ with $e\in
E(r_{2})$. ∎

######  Lemma 8.5.

For the coefficients of $\sigma(f^{+}_{\overline{w},3})\in E\langle
r_{1},r_{2}\rangle\\{(b_{*})_{3}\\}$ the system of Lemma 8.2 has for all $m>0$
no solution in $E\langle r_{1},r_{2}\rangle$, i.e. the assumptions of the
lemma are satisfied.

###### Proof.

In the expansion of $f((b_{*})_{3})=\sigma(f^{+}_{\overline{w},3})$ in the
statement of Lemma 8.2 only the coefficient $c_{0}$ contains a term with a
proper derivative which is $r_{1}r_{2}^{\prime}$.

First, we prove that for every $m$ there is no solution
$(\overline{y}_{0},\dots,\overline{y}_{m-1})\in E^{m}$. We consider the last
equation of the system whose only constant term (with respect to $y_{0}$, …,
$y_{m-1}$) is $mc_{0}$. We argue that the term $r_{1}r_{2}^{\prime}$ that is
thus introduced through $c_{0}$ into the last equation does not cancel. In
fact, since the coefficient of $r_{1}r_{2}^{\prime}$ is the constant $-1$ and
$r_{1}$, $r_{2}$ are differentially algebraically independent over $E$,
substitution of a putative solution
$(\overline{y}_{0},\dots,\overline{y}_{m-1})\in E^{m}$ does not lead to a
cancellation of $r_{1}r_{2}^{\prime}$.

Next we prove that for every $m>0$ the system has no solution
$(\overline{y}_{0},\dots,\overline{y}_{m-1})$ in $E\langle
r_{1},r_{2}\rangle^{m}$ with the property that at least one $\overline{y}_{i}$
satisfies $\overline{y}_{i}\notin E\langle r_{1}\rangle(r_{2})$. Suppose there
is such a solution and write every $\overline{y}_{i}$ as a completely reduced
fraction $a_{i}/b_{i}$ with $a_{i}$, $b_{i}\in E\\{r_{1},r_{2}\\}$, $b_{i}\neq
0$. We consider then the orders of all $a_{i}$, $b_{i}$ in the differential
indeterminate $r_{2}$ and choose an index $k$ where either $a_{k}$ or $b_{k}$
has highest order $d$. Then the assumption on the solution implies $d\geq 1$.
We substitute the solution into the system and consider the $k$-th equation.
We clear the denominator on the left hand side by multiplying both sides by
$b_{k}^{2}$, which yields the left hand side
$a_{k}^{\prime}b_{k}-a_{k}b_{k}^{\prime}$. The assumption that $a_{k}/b_{k}$
is completely reduced implies

$\frac{a_{k}}{b_{k}}\notin E\langle r_{1}\rangle(r_{2},\dots,r_{2}^{(d-1)})$

and so Lemma 8.3 applied with $K=E\langle r_{1}\rangle$ yields that the left
hand side has order $d+1$ in $r_{2}$. Since the maximal order in $r_{2}$ among
the coefficients on the right hand side is one and the maximal order among the
$a_{i}$, $b_{i}$ is $d\geq 1$, we conclude that the order in $r_{2}$ of the
right hand side is at most $d\geq 1$ while the left hand side has order $d+1$,
which implies that such a solution does not exist.

Proceeding in an analogous way with the roles of $r_{1}$ and $r_{2}$
exchanged, we conclude that no solution
$(\overline{y}_{0},\dots,\overline{y}_{m-1})$ in $E\langle
r_{1},r_{2}\rangle^{m}$ exists where at least one $\overline{y}_{i}$ satisfies
$\overline{y}_{i}\notin E\langle r_{2}\rangle(r_{1})$, where the contradiction
is derived by observing that the right hand side has order at most $d\geq 1$
in $r_{1}$, whereas the left hand side has order $d+1$ in $r_{1}$.

Finally we prove that for every $m\geq 1$ there is no solution
$(\overline{y}_{0},\dots,\overline{y}_{m-1})\in E(r_{1},r_{2})^{m}$ of
$\Sigma_{m}$. Suppose that for some $m$ there is one and assume that each
$\overline{y}_{i}=a_{i}/b_{i}$ is completely reduced. We consider the last
equation of the system. After substitution of
$(\overline{y}_{0},\dots,\overline{y}_{m-1})$ into the equation we again clear
the denominator on the left hand side by multiplying through by $b_{m-1}^{2}$.
The term $r_{1}r_{2}^{\prime}$ on the right hand side does not cancel because
all other terms are elements of $E[r_{1},r_{2}]$. Moreover, the indeterminate
$r_{1}^{\prime}$ does not occur on the right hand side, so that the left hand
side $a^{\prime}_{m-1}b_{m-1}-a_{m-1}b^{\prime}_{m-1}$ is independent of
$r_{1}^{\prime}$, too. By Lemma 8.4 this is the case if and only if
$a_{m-1}/b_{m-1}\in E(r_{2})$. This forces the left hand side to be in
$E\\{r_{2}\\}$, but the right hand side contains the term
$r_{1}r_{2}^{\prime}$. ∎

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