Universiteit Leiden Mathematisch Instituut
Geometry Seminar

Tuesday, 6 September 2005, 11:00 - 12:00, lecture room 312 of the Mathematisch Instituut Leiden.

Chris Peters (Grenoble): Can one effectively decide solvability of systems of differential equations?

Abstract:

A (complex-analytic) system of PDE is "generically" solvable by the Ansatz of a formal power series. To do this effectively, one has to perform the process of "prolongation". This consists of adding new equations in a way dictated by the original system. Usually this process has to be repeated a number of times. To find out how many times, one makes a linear approximation of the system through the "symbol" of the system. This symbol can be viewed as a homogeneous ideal in a polynomial algebra. In favorable situations the Castelnuovo-Mumford regularity of this ideal is the number of times one has to prolong the system. To calculate it effectively a new method will be discussed based on Groebner basis and effective homology from algebraic topology to calculate the regularity (and much more). Computer programs implementing this which give very fast results are Macaulay and Kenzo. I intend to explain all this first for a system of linear PDE, and then, if time permits, explain the geometric approach to PDE needed for the general case. This is a report on work in progress of the Grenoble group of the European project "GIFT"