Home

The Kolchin Seminar in Differential Algebra

Hunter College - Room HW706

Saturday December 13th 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.

******************************************

Jerry Kovacic
City College - CUNY
Hyperelliptic Jacobians in differential Galois theory. A preliminary report
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.

******************************************

Felix Ulmer
Université de Rennes
Construction of linear differential equations for finite groups
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.

******************************************

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

Click here for directions to Hunter College and location of room

Home