168 research outputs found
A Reduced Form for Linear Differential Systems and its Application to Integrability of Hamiltonian Systems
Let with be a differential linear
system. We say that a matrix is a {\em reduced
form} of if and there exists such that . Such a form is
often the sparsest possible attainable through gauge transformations without
introducing new transcendants. In this article, we discuss how to compute
reduced forms of some symplectic differential systems, arising as variational
equations of hamiltonian systems. We use this to give an effective form of the
Morales-Ramis theorem on (non)-integrability of Hamiltonian systems.Comment: 28 page
Galois theory on the line in nonzero characteristic
The author surveys Galois theory of function fields with non-zero
caracteristic and its relation to the structure of finite permutation groups
and matrix groups.Comment: 66 pages. Abstract added in migration
Mazur-Tate elements of non-ordinary modular forms
We establish formulae for the Iwasawa invariants of Mazur--Tate elements of
cuspidal eigenforms, generalizing known results in weight 2. Our first theorem
deals with forms of "medium" weight, and our second deals with forms of small
slope . We give examples illustrating the strange behavior which can occur in
the high weight, high slope case
Hrushovski's Algorithm for Computing the Galois Group of a Linear Differential Equation
We present a detailed and simplified version of Hrushovski's algorithm that
determines the Galois group of a linear differential equation. There are three
major ingredients in this algorithm. The first is to look for a degree bound
for proto-Galois groups, which enables one to compute one of them. The second
is to determine the identity component of the Galois group that is the pullback
of a torus to the proto-Galois group. The third is to recover the Galois group
from its identity component and a finite Galois group.Comment: 27 page
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