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# A modern perspective on Fano’s approach to linear differential equations

17 October 2008

PhD ceremony: K.A. Nguyen, 13.15 uur, Academiegebouw, Broerstraat 5, Groningen

Thesis: A modern perspective on Fano’s approach to linear differential equations

Promotor(s): prof. M. van der Put, prof. J. Top

Faculty: Mathematics and Natural Sciences

This thesis is concerned with three different main themes. First of all we focus on the following question, originally posed by L. Fuchs (1883): Is it possible to obtain all solutions of a given linear differential equation of order n by solving a sequence of homogeneous differential equations of order d (for some 1 d < n) and algebraic extensions, under the assumption that the n independent solutions satisfy a nontrivial homogeneous equation over C?

We treat this problem in Chapter 1 and obtain the main result (Theorem 1.2.20) saying that the minimal such d can be derived from the differential Galois group in an explicit way. Chapters 2 and 3 are devoted to “solve” an absolutely irreducible differential equation explicitly in terms of equations of lower order (whenever possible). This problem is formulated in terms of differential modules. In Chapter 1, criteria are given under which this is possible. In Chapter 2 we present some necessary background on Lie algebras. This is used in Chapter 3 to actually “solve” various types of differential modules. The results extend the classical work of G. Fano.

Secondly, in Chapter 4 we compare the differential Galois group of a linear differential equation with a similar group introduced by G. Fano in 1900.

Finally, every differential Galois group is an algebraic subgroup of GLn(C) and it is well-known that each algebraic subgroup occurs as a differential Galois group over the field C(z). Chapter 5 and Appendix A present a complete classification of the algebraic subgroups of GL2(C). In Chapter 6, we produce for most of these groups a corresponding second order differential equation.

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