Abstract:
This is an expository article describing some applications of
parameterized Picard–Vessiot theory. This Galois theory for parameterized
linear differential equations was Cassidy and Singer's contribution to an
earlier volume dedicated to the memory of Andrey Bolibrukh.
The main results we present here were obtained for families
of ordinary differential equations with parameterized regular singularities
in joint work with Singer.
They include parametric versions of Schlesinger's theorem and
of the weak Riemann–Hilbert problem as well as an algebraic characterization
of a special type of monodromy evolving deformations illustrated by the
classical Darboux–Halphen equation. Some of these results have recently
been applied by different authors to solve the inverse problem of
parameterized Picard–Vessiot theory, and were also generalized to irregular
singularities. We sketch some of these results by other authors. The paper
includes a brief history of the Darboux–Halphen equation as well
as an appendix on differentially closed fields.
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