Mathematics

My research interests are polynomial mappings and locally nilpotent derivations, or, from a more geometrical viewpoint, regular maps from Cn to Cm and algebraic actions of (C,+) on Cn. The most famous problem in the area of polynomial mappings is the Jacobian Conjecture: every polynomial map from Cn to Cn with determinant of the Jacobian matrix equal to 1 is invertible.

Locally nilpotent derivations are the main subject of my PhD-thesis. A well-known problem here is the Cancellation Problem: does the fact that A[T] is isomorphic to C[X1,...,Xn] imply that A itself is isomorphic to C[X1,...,Xn-1]? This problem can be reformulated in terms of locally nilpotent derivations. For n at most 3, the problem has been solved affirmatively; together with Arno van den Essen(U. Nijmegen) and Harm Derksen (U. Michigan) I have shown a partial result for n=4; for all other n the problem is still completely open. From a geometrical point, locally nilpotent derivations can also be seen as (C,+)- actions on Cn. My research at New Mexico State University focusses on the ring of invariants of locally trivial and of proper (C,+)-actions. The ultimate hope is to show (or disprove) that every proper (C,+)-action on C4 is in fact trivial.

My research often involves finding concrete examples. Therefore, I am also very much interested in computational aspects of algebraic geometry, which has lead to the Jacobian Package.

Jacobian Package

The Jacobian Package is a collection of Maple routines for dealing with explicit computations in the area of polynomial mappings and locally nilpotent derivations. It has been developed at University of Nijmegen, by various people including myself. I am currently maintaining an expanded version of this package written in Maple, Magma, and Singular. While all computations can be done using Maple alone, several routines have alternative, faster implementations in Magma and/or Singluar.

Publications