Conversion Between Implicit and Parametric Representation of Differential Varieties

Xiao-Shan Gao, Institute of Systems Science, Chinese Academy of Sciences.

October 6th, 2004, at 3:30pm in K9509.

We will first give an introduction to the characteristic set method for
algebraic differential equations, which could serve a basic tool for
constructive algebraic geometry and provides a decision procedure for 
the first order theory of differential closed fields.  As applications, 
we will present some recent results on conversion between implicit 
representation and parametric representation of differential varieties.

For a set of differential rational parametric equations, we show how to 
compute its image and decide whether it represents a rational differential 
variety.  The parametrization problem is much more difficult and is solved only
in very limited cases. We give a polynomial time algorithm to find the rational
parametrization for an autonomous first order ODE. Algorithms to find algebraic
solutions for autonomous first order ODEs will also be discussed.