Parameter Differential Equations for the Classical Groups and Their Generic Properties

In this talk we compute explicit and nice linear differential equations which have the classical groups as differential Galois groups.

The realization of these groups is done over a differential field which is purely differential transcendental over its constants and so the coefficients of our equations will be differential polynomials in a specific number of differential indeterminates.

We explain further how we use the geometric structure of the groups for the construction of our equations.

In a second part of this talk we discuss the generic properties of our linear parameter differential equations and the related problems. In case of the special linear group and the exceptional group of type G2 it turns out that our equations are generic for algebraically closed differential fields.