Exponents, Levelt lattices and Fuchsian relations for regular differential systems.
(Exposants, résaux de Levelt et relations de Fuchs pour les systèmes différentiels réguliers.)

*(French)*Zbl 1025.34090
Prépublication de l’Institut de Recherche Mathématique Avancée. 39. Strasbourg: Université Strasbourg, Institut de Recherche Mathématique Avancée, 84 p. (1999).

The main goal of this thesis is to obtain a Fuchs relation for regular differential systems. To obtain this result, the author gives a new definition, an algebraic one, for the exponents of the system. This problem is well known for a linear differential equation. Indeed, if \(Ly=0\) is a linear differential equation of order \(n\) with coefficients in the field \(K= \mathbb{C}\{z\}[z^{-1}]\) of convergent Laurent series, then the exponents \(e^s_i\), \(1\leq i\leq n\), at a singular point \(s\), characterize the growth of solutions in the neighborhood of \(s\) and these exponents are easily calculated from the equation. Moreover, for such an equation we have the famous Fuchs relation:
\[
\sum_{s\in P^1(\mathbb{C})} \left(\sum^n_{i=1} e^s_i-{n(n-1)\over 2} \right)= -n(n-1).
\]
On the other hand, for a regular differential system \(Y'=A Y\), with \(A\in\text{Mat}(n,K)\), we have an analytic definition of the exponent due to A. H. M. Levelt [Nederl. Akad. Wet., Proc., Ser. A 64, 373-385 (1961; Zbl 0124.03602)]. However, before this thesis we did not know an analog of the Fuchs relation, except for the case where \(A\) has only simple poles. Indeed, if \(A=A_{-1}/z+A_0 +zA_1+z^2A_2+\), where the \(A_i\)’s are constant matrices, F. R. Gantmacher [The theory of matrices (Russian) (Moskau: Staatsverlag für technisch-theoretische Literatur (1953; Zbl 0050.24804)] gave an algebraic definition or the exponents \(e^0_i\) of the system: they are the eigenvalues of \(A_{-1}\). Also, A. A. Bolibrukh [The 21st Hilbert problem for linear Fuchsian systems. (Russian) Trudy Matematicheskogo Instituta Imeni V. A. Steklova. Moskva: Nauka (1994; Zbl 0844.34003), Proposition 1.2.3)] proved that if all the poles of \(A\) are simple, then we have the relation \(\sum_{s\in P^1(\mathbb{C})} \sum^n_{i=1} e^s_i=0\). But in the general case, the only result known was \(\sum_{s\in P^1(\mathbb{C})} \sum^n_{i=1} e^s_i\leq 0\) [A. A. Bolibrukh, loc. cit].

To improve this inequality, the author first gives an algebraic definition of the exponents. Let \(O=\mathbb{C}\{z\}\) be the ring of the holomorphic functions in the neighborhood of \(0\), and let \((e)\) be the canonical basis of \(K^n\). Let \(Oe\) be the lattice generated by the basis \((e)\), i.e., the \(O\)-module generated by \((e)\). If 0 is a regular singular point for the differential system, we know that there exists a gauge transformation \(Y= PZ\), with \(P\in\text{GL}(n,K)\) such that the new system \(Z'=A_{[P]}Z\) has a simple pole or, equivalently, that the lattice \(Oe'\), where \(e'=(e)P\) is the basis of \(K^n\) defined by the matrix \(P\), is stable under the connection \(\nabla_{zd/dz}\) associated with the differential system. When we have a differential system with a simple pole (or, equivalently, a stable lattice) we can use Gantmacher’s algebraic definition of exponents. But the problem is that, in general, different stable lattices give different values to the exponents! Then the author defines a particular lattice: the Levelt lattice defined as the biggest stable lattice included in the original lattice \(Oe\). The (algebraic) exponents are then defined with this lattice. The author shows that these algebraic exponents are the same as Levelt’s analytic exponents.

The advantages of this point of view are various. The author gives a simple and intrinsic definition of the exponents based on lattice theory. Using elementary properties of this theory, the author characterizes the Levelt lattice as the “closest” stable sublattice of the original lattice. After that, using both stability and maximality of the Levelt lattice, he obtains in a very elegant way the following Fuchs relation for a regular differential system \[ -{n(n-1)\over 2}h(A)\leq\sum_{s\in P^1(\mathbb{C})}\sum^n_{i=1}e^s_i\leq -h(A), \] where \(h(A)= \sum_{a\in P^1(\mathbb{C})} \sup(0,-\text{ord}_aAdz-1)\) and \(\text{ord}_a\) is the order at \(a\).

{Reviewer’s remarks: (1) The author proves that this relation is optimal. There exist systems for which the displayed sum equals \(-h(A)\), and others for which \(\sum_{s\in P^1(\mathbb{C})} \sum^n_{i=1} e^s_i=-{n(n-1) \over 2}h(A)\). (2) We have that \(h(A)=0\) iff \(A\) has only simple poles, and in this case we obtain the previous result of Bolibrukh.}

In conclusion, the results of this thesis unify and generalize the previous results of Gantmacher, Levelt and Bolibrukh in this area, and should give way to an effective algorithm for the calculation of the exponents of a differential system.

To improve this inequality, the author first gives an algebraic definition of the exponents. Let \(O=\mathbb{C}\{z\}\) be the ring of the holomorphic functions in the neighborhood of \(0\), and let \((e)\) be the canonical basis of \(K^n\). Let \(Oe\) be the lattice generated by the basis \((e)\), i.e., the \(O\)-module generated by \((e)\). If 0 is a regular singular point for the differential system, we know that there exists a gauge transformation \(Y= PZ\), with \(P\in\text{GL}(n,K)\) such that the new system \(Z'=A_{[P]}Z\) has a simple pole or, equivalently, that the lattice \(Oe'\), where \(e'=(e)P\) is the basis of \(K^n\) defined by the matrix \(P\), is stable under the connection \(\nabla_{zd/dz}\) associated with the differential system. When we have a differential system with a simple pole (or, equivalently, a stable lattice) we can use Gantmacher’s algebraic definition of exponents. But the problem is that, in general, different stable lattices give different values to the exponents! Then the author defines a particular lattice: the Levelt lattice defined as the biggest stable lattice included in the original lattice \(Oe\). The (algebraic) exponents are then defined with this lattice. The author shows that these algebraic exponents are the same as Levelt’s analytic exponents.

The advantages of this point of view are various. The author gives a simple and intrinsic definition of the exponents based on lattice theory. Using elementary properties of this theory, the author characterizes the Levelt lattice as the “closest” stable sublattice of the original lattice. After that, using both stability and maximality of the Levelt lattice, he obtains in a very elegant way the following Fuchs relation for a regular differential system \[ -{n(n-1)\over 2}h(A)\leq\sum_{s\in P^1(\mathbb{C})}\sum^n_{i=1}e^s_i\leq -h(A), \] where \(h(A)= \sum_{a\in P^1(\mathbb{C})} \sup(0,-\text{ord}_aAdz-1)\) and \(\text{ord}_a\) is the order at \(a\).

{Reviewer’s remarks: (1) The author proves that this relation is optimal. There exist systems for which the displayed sum equals \(-h(A)\), and others for which \(\sum_{s\in P^1(\mathbb{C})} \sum^n_{i=1} e^s_i=-{n(n-1) \over 2}h(A)\). (2) We have that \(h(A)=0\) iff \(A\) has only simple poles, and in this case we obtain the previous result of Bolibrukh.}

In conclusion, the results of this thesis unify and generalize the previous results of Gantmacher, Levelt and Bolibrukh in this area, and should give way to an effective algorithm for the calculation of the exponents of a differential system.

Reviewer: Élie Compoint (MR 2001f:34173)

##### MSC:

34M35 | Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms |

34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |

34A30 | Linear ordinary differential equations and systems, general |

34-04 | Software, source code, etc. for problems pertaining to ordinary differential equations |

12H05 | Differential algebra |

32S40 | Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects) |