**The Kolchin Seminar in Differential
Algebra **

**Saturday December 13 ^{th} 2003**

**Felix Ulmer**

Université de Rennes, France**Algorithms for computing Liouvillian solutions**

2:00 - 3:00PM

**Abstract:** I plan to discuss bounds for the logarithmic derivative of a solution of a linear differential equation (in Characteristic zero) and show how to
compute the minimal polynomial of such a logarithmic derivative (i.e. an
algebraic solution of the Riccati equation associated to the linear
differential equation). I will therefore discuss generalizations of the
Kovacic algorithm to equations of order higher than two, in particular
to third order equations.

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**Jerry Kovacic**City College - CUNY

3:30 - 4:30PM

**Abstract:** Jacobians of hyperelliptic curves have been studied
recently for their application to cryptography and the
KdV equations. In the first case one works over
a finite field, in the second case one works in analysis.
I work in characteristic 0 and pure algebra. Nonetheless,
the hyperelliptic Jacobians are still interesting. They
supply examples of differential Galois extensions that
are not Picard-Vessiot.We will review the algebraic theory using the characterization
of Mumford (Tata lectures on Theta II). There are very efficient
algorithms available for performing arithmetic in the Jacobian,
and these have been used for a Risch-type algorithm which
computes hyperelliptic integrals. But I don't think there are
any algorithms for linear homogeneous differential
equations. I recommend this as a ripe area for research.
For differential Galois theory, we are interested in the
logarithmic derivative, a certain homomorphism
from the Jacobian to its Lie algebra. It is also a morphism in the
category of differential varieties. We are able to write down explicit
formulas, which we can use to construct a strongly normal extension by
"integration". This is analogous to how a Picard-Vessiot extension
is formed. In our case the Galois group is a subgroup of the
hyperelliptic Jacobian.
An explicit example can be obtained using the Klein
sigma-function and Weierstrass p-functions of Baker,
Buchstaber, Enolskii, et. al.

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**Felix Ulmer**Université de Rennes

5:00 - 6:00 PM

**Abstract:** This is a joint work with Marius van der Put that appeared in
Journal of Algebra, 226, 2000, 920-966. We generalize a method due to
Hurwitz to construct a linear differential equation for a given
representation of a finite group. The method is quite efficient for
lower order and I would like show the method and illustrate its
limitations.

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**Coffee and tea will be served beginning at 1:30 PM Room
HW706 **

**Click
here for directions to Hunter College and location of room **