Liouvillian solutions of linear differential equations of order three and higher
This paper contains several results of a different kind,
all of which are necessary ingredients for
efficient computation of Liouvillian solutions of equations of order 3.
The first new result is theorem 2.1, which shows how the
Liouvillian solutions can be obtained directly from the
(semi)-invariants. This eliminates the main bottleneck in
After that it is shown how to compute semi-invariants
while avoiding another bottleneck (the computation of
We then show for order 3 how one can avoid
computing semi-invariants as much as possible (getting
the necessary information from invariants, which can
be computed more easily) using constructions with
(semi)-invariants (section 2.3), and by studying each
group separately (section 4.3). This is yet another
big efficiency improvement. Furthermore by this list
of groups we know the degrees of the invariants we need
to compute, and we can avoid another bottleneck (the
problem that the appendix deals with) in most cases.
In the appendix we treat the problem of how to find values
of parameters for which polynomials can be factored.
There is a partial implementation (joint work with J.A. Weil) which
we used to compute examples, but we never shaped up the code to the
point where it can be distributed. However,
there is now also a newer implementation
that is available for download but it only treats the imprimitive case.
Bronstein has written a complete implementation for order 3
that can be used online through his webpage.
Click here for verifying the
correctness of the example in section 4.3.10. Here is the code for
obtaining the operator for the FP28 example
of order 4 and for verifying the Riccati polynomial in our paper. Note that
the last two words "and higher" should be removed from the title of the
paper because although the method in the paper works very well for order
3, it is not the best approach for higher order (see also the paper by
Olivier Cormier in ISSAC'2001).
Here is another nice example, with Galois
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