Differential Galois Theory and Integration

We consider two confluent Heun²²2See https://dlmf.nist.gov/31.12 or Mot (18). The notation $HeunC$ is the syntax in Maple. functions

and

They form a basis of solutions of the second order linear differential equation

The Wronskian relation gives us the algebraic relation $(f_{1}f_{2}^{\prime}-f_{1}^{\prime}f_{2})^{3}=x^{2}(x-1)^{2}$. The equation has order two so the Kovacic algorithm Kov (86); UW (96); vHW (05) can be used to compute the differential Galois group and we find that no other algebraic relations exist between $f_{1}$, $f_{2}$, $f_{1}^{\prime}$ and $f_{2}^{\prime}$. Now let $F_{i}(x):=\int^{x}f_{i}(t)dt$ be a primitive. We want to determine whether $F_{1}$ and $F_{2}$ are algebraically independent of the $f_{i}$ and $f_{i}^{\prime}$ or not. The techniques explained below will show that this question reduces to asking whether there is a rational solution to the linear differential system

or, equivalently, whether the following linear differential equation (the left hand side turns out to be the adjoint operator of $L$) has a rational solution:

It can be seen directly or with a computer algebra system that this equation has no rational solution. The underlying theoretical tools are from the constructive differential Galois theory. However, in operational terms, it is rather easy to compute and check : no hard theory is required for the calculation. Let us unveil a corner of the underlying tools.

We have in fact studied the differential Galois group of the differential system $[A]$ with

which admits the fundamental solution matrix

Our calculation of rational solution above shows (with the tools displayed below) that the differential Galois group of the system $[A]$ will have dimension $5$. This in turn shows that the integrals $\int^{x}f_{j}(t){\rm dt}$ are algebraically independent of $f_{1}$, $f_{2}$ and their derivatives. In fact, it even shows that both integrals are algebraically independent.

Recall that the hypergeometric function is given by the formula

We now take two hypergeometric functions (with nice modular properties)

and

The vectors $Y_{i}:=(f_{i},f_{i}^{\prime})^{T}$ are solutions of $Y^{\prime}=A_{1}Y$ below. To study properties of their integrals, we set $Y:=(f,f^{\prime},\int^{x}f(t)dt)^{T}$ and we have $Y^{\prime}=AY$ where

We will see in the sequel how we may find a suitable change of variables

such that

This shows that

and the differential Galois group of $[A]$ has dimension 3. We note that none of these two examples is new and that efficient methods to handle these questions have been developed by Abramov and van Hoeij in AvH (99, 97).

These two examples have shown how, by augmenting the dimension of a linear differential system, we can study integrals of its solutions. Conversely, block triangular systems give rise to integrals via variation of constants; we review this for clarification. For a factorized reducible system $[A]$ of the form

with $A_{diag}:=\left(\begin{array}[]{c|c}A_{1}&0\\ \hline\cr 0&A_{2}\end{array}\right)$ and $A_{sub}:=\left(\begin{array}[]{c|c}0&0\\ \hline\cr S&0\end{array}\right)$, we have a fundamental solution matrix of the form

Once $U_{1}$ and $U_{2}$ are known, $V$ is given by integrals : $\partial V=U_{2}^{-1}SU_{1}$. So it may seem that no further theory is required. However, $U_{1}$ and $U_{2}$ are fundamental solution matrices for systems $\partial U_{i}=A_{i}U_{i}$ so the relation $\partial V=U_{2}^{-1}SU_{1}$ may involve integrals of complicated $D$-finite functions.
Our approach will be to first “reduce” $S$ as much as possible, using manipulations with rational functions, to prepare the system for easier solving. In return, we will obtain algebraic information on all (possibly iterated) integrals that may occur in the relation $\partial V=U_{2}^{-1}SU_{1}$.

We note, for the record, that factorized differential operators also correspond to block triangular differential systems.

In what follows, $(\mathbf{k},\partial)$ is a differential field of characteristic $0$. We outline a brief exposition of the Galois theory of linear differential equations, see PS (03); CH (11); Sin (09) for expositions with proofs. We consider a linear differential system

Let ${\cal C}$ be the field of constants of the differential field $\mathbf{k}$, that is ${\cal C}:=\{a\in\mathbf{k}|\partial(a)=0\}$. We will assume that ${\cal C}$ is algebraically closed, i.e, every non constant polynomial equation has a solution in ${\cal C}$.

A Picard-Vessiot extension is a field $K=\mathbf{k}(U)$, where $U$ is a fundamental solution matrix³³3An invertible $n\times n$ matrix $U$ such that $U^{\prime}=AU$. of $[A]$, such that the field of constants of $K$ is still ${\cal C}$. This can be constructed algebraically; alternatively, when $\mathbf{k}$ is a field of meromorphic functions, one may consider a local matrix of power series solutions at a regular point and this gives a Picard-Vessiot extension. A Picard-Vessiot extension is unique modulo differential field isomorphisms.

The differential Galois group $G:=\mathrm{Aut}_{\partial}(K/\mathbf{k})$ is the set of automorphisms of $K$ which leave the base field $\mathbf{k}$ fixed and commute with the derivation. Let $\sigma\in G$. By construction, $\sigma(U)$ is also a fundamental solution matrix in $K$ and we find that there exists a matrix $[\sigma]\in\mathrm{GL}(n,{\cal C})$ such that $\sigma(U)=U\cdot[\sigma]$. The map $\sigma\mapsto[\sigma]$ provides a faithful representation of $G$ as a subgroup of $\mathrm{GL}(n,{\cal C})$, actually a linear algebraic group. If we change the fundamental solution, we obtain a conjugate representation.

We recall that, given an invertible matrix $P\in\mathrm{GL}(n,\mathbf{k})$, the linear change of variables $Y=PZ$ produces a new differential system denoted $\partial Z=P[A]Z$ where

We note that such a gauge transformation $P[A]$, with $P\in\mathrm{GL}(n,\mathbf{k})$, does not change the Galois group.

Given a Picard-Vessiot extension $K=\mathbf{k}(U)$, the polynomial relations among all entries of $U$ (over $\mathbf{k}$) form an ideal $I$. The Galois group $G$ can then be viewed as the set of matrices which stabilize this ideal $I$ of relations. Thus the computation of $G$ is strongly related to the understanding of the algebraic relations among the solutions.

The Galois-Lie algebra of $[A]$ is the Lie algebra $\mathfrak{g}$ of the differential Galois group $G$. It is defined as the tangent space of $G$ at the identity $\mathrm{Id}$. The dimension of $\mathfrak{g}$ measures the transcendence degree of $K$ over $\mathbf{k}$, that is

One way of computing the Lie algebra (PS (03)) is the following : $\mathfrak{g}$ is the set of matrices $N$ such that $\mathrm{Id}+\epsilon N\in G({\cal C}[\epsilon])$ with $\epsilon^{2}=0$. In other terms, $\mathrm{Id}+\epsilon N$ satisfies the defining equations of the group modulo $\epsilon^{2}$. For example, ${\mathrm{SL}(n,{\cal C})}$ (set of $M$ such that $\det(M)=1$) gives the Lie algebra $\mathfrak{sl}(n,{\cal C})$ of matrices $N$ such that $\mathrm{Tr}(N)=0$. The symplectic group ${\mathrm{Sp}(2n,{\cal C})}$ is the set of $M$ such that $M^{T}\cdot J\cdot M=J$; its Lie algebra $\mathfrak{sp}(n,{\cal C})$ is found to the set of $N$ such that $N^{T}J+JN=0$, with $J=\left(\begin{array}[]{cc}0&\mathrm{Id}\\ -\mathrm{Id}&0\end{array}\right)$. The additive group $\mathbb{G}_{a}:=\left\{\left(\begin{array}[]{cc}1&a\\ 0&1\end{array}\right),a\in{\cal C}\right\}$ admits the Lie algebra ${\mathfrak{g}_{a}}=Span_{{\cal C}}\left\{\left(\begin{array}[]{cc}0&1\\ 0&0\end{array}\right)\right\}$; the multiplicative group $\mathbb{G}_{m}=\left\{\left(\begin{array}[]{cc}a&0\\ 0&\frac{1}{a}\end{array}\right),a\in{\cal C}^{*}\right\}$ admits the Lie algebra $\mathfrak{g}_{m}=Span_{{\cal C}}\left\{\left(\begin{array}[]{cc}1&0\\ 0&-1\end{array}\right)\right\}$.

The reduction technique exposed in this chapter aims at computing directly the Galois-Lie algebra $\mathfrak{g}$ before computing $G$ itself. Although the theory is not obvious, the resulting calculations are reasonably simple.

The Lie algebra $\textrm{Lie}(A)$ associated to the matrix $A$ is defined as follows. Let $a_{1},\ldots,a_{s}\in\mathbf{k}$ be a basis of the ${\cal C}$-vector space generated by the coefficients of $A$. We can then decompose $A$ as $A=\sum_{i=1}^{s}a_{i}M_{i}$ where the $M_{i}$ are constant matrices. Now, we consider the smallest Lie algebra containing all the $M_{i}$: this is the vector space generated by the $M_{i}$ and all their iterated Lie brackets ($[M,N]:=MN-NM$); then we take its algebraic envelope.

The decomposition $A=\sum_{i=1}^{s}a_{i}M_{i}$ is not unique but the vector space generated by the $M_{i}$ is unique. Thus, the associated Lie algebra $\textrm{Lie}(A)$ does not depend on the chosen decomposition.
This Lie algebra $\textrm{Lie}(A)$ appears in works of Magnus Mag (54) or Feynman who use the Baker-Campbell-Hausdorf formula to write solutions of $\partial Y=AY$ as (infinite) products of exponentials constructed with Lie brackets. Wei and Norman give in WN (63, 64) a finite formula to solve the system when $\textrm{Lie}(A)$ is solvable. This formula is well-known in physics and control theory but not as well among mathematicians. The terminology of Lie algebra associated to $A$ appears⁵⁵5For this reason, some authors, including ourselves, call the decomposition $A=\sum_{i=1}^{s}a_{i}M_{i}$ a Wei-Norman decomposition of $A$. in WN (63, 64) (in there, it is defined as the Lie algebra generated by all values of $A(z_{0})$ for $z_{0}$ spanning all constants minus singularities and the algebraic envelope is missing). In the sequel, we will study a $4$-dimensional example and an $8$-dimensional example where our technique will have some relations to the Wei-Norman approach; namely, we will change $A$ to obtain an associated Lie algebra $\textrm{Lie}(A)$ of minimal dimension so that solving formulas become optimal in some sense.

We now turn to the link between $\textrm{Lie}(A)$ and differential Galois theory, based on two important results of Kolchin and Kovacic. Proofs can be found in PS (03), Proposition 1.31 and Corollary 1.32; see also BSMR (10), Theorem 5.8, and AMCW (13), § 5.3 after Remark 31. Let $A\in\mathrm{Mat}(n,\mathbf{k})$; let $G$ be the differential Galois group of $[A]$ and $\mathfrak{g}$ its Lie algebra, the Galois-Lie algebra of the system $[A]$. The first result is

So, the Lie algebra associated to $A$, an object which is very easy to compute, provides an “upper bound” on $\mathfrak{g}$. When we perform a gauge transformation $P\in\mathrm{GL}(n,\mathbf{k})$ to obtain a new system $P[A]$, $G$ and $\mathfrak{g}$ are invariant while $\textrm{Lie}(P[A])$ may vary. This shows that $\mathfrak{g}$ is a lower bound on all $\textrm{Lie}(P[A])$, for all gauge transformations $P\in\mathrm{GL}(n,\mathbf{k})$. The second result of Kolchin and Kovacic is that this lower bound is reached. By definition, $\textrm{Lie}(A)$ is the Lie algebra of an algebraic connected group $H$. Then there exists a gauge transformation⁶⁶6The notation $H(\overline{\mathbf{k}})$ denotes matrices whose entries are in $\overline{\mathbf{k}}$ and satisfy all the equations defining the algebraic group $H$. $P\in H(\overline{\mathbf{k}})$ such that $\mathfrak{g}=\textrm{Lie}(P[A])$. Furthermore, if $G$ is connected and under the very mild additional condition that $\mathbf{k}$ is a $\mathcal{C}^{1}$-field⁷⁷7 A field $\mathbf{k}$ is a $\mathcal{C}^{1}$-field when every non-constant homogeneous polynomial $P$ over $\mathbf{k}$ has a non-trivial zero provided that the number of its variables is more than its degree. For example, ${\cal C}(x)$ is a $\mathcal{C}^{1}$-field and any algebraic extension of a $\mathcal{C}^{1}$-field is a $\mathcal{C}^{1}$-field (Tsen’s theorem). then we may choose $P\in H({\mathbf{k}})$ (no algebraic extension).

The results of Kolchin and Kovacic show that a reduced form always exists. We provide, in the sequel, constructive methods to obtain them when the systems are in block-triangular form. They will be illustrated on our $4$-dimensional example in the next section.

The ideas behind this notion of reduced form have been used for inverse problems in differential Galois theory: given an algebraic group $G$, construct a differential system $[A]$ having $G$ as its differential Galois group. It is also a technique known in differential geometry. Its use for direct problems in differential Galois theory is more recent. A remark in PS (03) suggests that this would be a good idea. In the context of Lie-Vessiot systems, Blazquez and Morales exploit this idea in BSMR (10). It is developed in AMDW (16); AMW (12, 11) in order to study variational equations in the context of integrability of Hamiltonian systems and the Morales-Ramis-Simó theory (and later in CW (18) to study algebraic properties of Painlevé equations). For irreducible systems (or systems in block diagonal form), a criterion for reduced forms is established in AMCW (13) with a decision procedure. Another, much more efficient approach is given in BCWDV (16); BCDVW (20) together with generalizations of the criterion of AMCW (13).
The approach described below allows, given the above results, to compute a reduced form of a block triangular linear differential system (the last case remaining after all the above contributions). It is based upon these works, notably CW (18), and is constructed in DW (19).

Let $\mathfrak{h}_{sub}$ be the the set of off-diagonal constant matrices of the form $\left(\begin{array}[]{c|c}0&0\\ \hline\cr{S}&0\end{array}\right)$ (same sizes as in relation (1)). We will extend the scalars to $\mathfrak{h}_{sub}(\mathbf{k}):=\mathfrak{h}_{sub}\otimes_{{\cal C}}\mathbf{k}$, the off-diagonal matrices with coefficients in $\mathbf{k}$. Our first step is that we may find a reduction matrix in a very particular shape.

The following is is based on an observation from AMW (12, 11). Let $P=\mathrm{Id}+B$, $B\in\mathfrak{h}_{sub}(\mathbf{k})$. Suppose that, for all $Q\in\left\{\mathrm{Id}+B,B\in\mathfrak{h}_{sub}(\mathbf{k})\right\}$, we have $\textrm{Lie}(P[A])\subseteq\textrm{Lie}(Q[P[A]])$; then, $P[A]$ is in reduced form. In other terms, no rational gauge transformation can turn it into a system with a smaller associated Lie algebra. In this case, $\textrm{Lie}(P[A])$ will be the Lie algebra of the differential Galois group and this will give us transcendence relations and algebraic relations on the solutions; this will be seen on the main example of this section.

More generally, as we can see in DW (19), Section 5, if our method can reduce a system with two diagonal blocks then we can iterate this method to obtain a reduced form of a block-triangular system with an arbitrary number of blocks on the diagonal. Let us illustrate this iteration on a system with three diagonal blocks of the form

where the block diagonal part is in reduced form (see BCDVW (20); BCWDV (16) for this). We will first reduce the south-east part (which is of the same form as (1)) into a form

Let $P_{1}$ be the reduction matrix. By DW (19), Lemma 5.1, the following system is automatically in reduced form

Now we perform the gauge transformation $\left(\begin{array}[]{c|c}\mathrm{Id}&0\\ \hline\cr 0&P_{1}\end{array}\right)$ to obtain a system of the form

(the $S_{i,j}$ may have changed after the first reduction step). We now see that this system in the same form as (1) with $A_{d}$ as the block diagonal matrix. So a second reduction of a two-blocks triangular system allows to reduce the initial three-blocks triangular system.

This iteration is well seen in our $4$-dimensional example below.

To summarize what this example suggests: we need to “cancel” terms in the purely triangular part; this reduces to finding rational solutions of linear differential equations with parametrized right-hand sides. And the order of the computations matters: here, one needs to study the relations on $f_{2}$ and $f_{3}$ before studying relations on $f_{1}$. We will show, in the sequel, how to systematize these ideas, using an isotypical decomposition and an adapted flag structure, and how to make them algorithmic so that a computer algebra system may perform the calculations.

In this example, we had seen that a fundamental solution matrix could be written using $e^{x},\ln(x),\ln(x-1)$ and dilog(x). As $[B]$ is in reduced form, $\textrm{Lie}(B)$ is the Lie algebra of the Galois group and it has dimension $4$. This shows that these four functions are transcendent and algebraically independent. So our calculation above (long but not hard) gave us a simple proof that dilog(x) is algebraically independent of $e^{x},\ln(x),\ln(x-1)$.

We recall our notations so far. We have $n_{i}\times n_{i}$ matrices $A_{i}$ with coefficients in $\mathbf{k}$ and

If we take two off-diagonal matrices $B_{1}$ and $B_{2}$ in $\mathfrak{h}_{sub}$, we have $B_{1}.B_{2}=0$. This allows two simple calculations. First, let $P:=\mathrm{Id}+\sum_{i}f_{i}B_{i}$, with ${f_{i}}\in\mathbf{k}$, $B_{i}\in\mathfrak{h}_{sub}$. Then

Furthermore, $[{{A_{\mathrm{diag}}},B_{i}}]\in\mathfrak{h}_{sub}(\mathbf{k})$. These two calculations show that reduction will be governed by the adjoint action ${\Psi}:X\mapsto[{{A_{\mathrm{diag}}},X}]$ of the block diagonal part ${A_{\mathrm{diag}}}$ on $\mathfrak{h}_{sub}(\mathbf{k})$. This adjoint action $\Psi$ is a linear map. Its matrix, on the canonical basis of $\mathfrak{h}_{sub}$, is