As a member of the research unit "Algebraic Geometry and Number Theory" of the Catholic University Leuven (Belgium), I prepare my doctoral thesis under supervision of Jan Denef. My research domain is:

Igusa's local zeta function & Newton polyhedra

In [1], Jan Denef and myself give a very explicit formula for Igusa's local zeta function associated to polynomials f, which are sufficiently non-degenerated with respect to its Newton polyhedron. The actors in this formula are objects of convex geometry (the Newtonpolyheder of f, its faces, the cones associated to the faces) and numbers of zero's of polynomials in F_p \ {0}, with F_p the finite field of p elements.

We are very much interested in the largest real pole of Igusa's local zeta function and its order, because it gives important information about the numbers of solutions of f(x) 0 mod pⁿ for n=1,2,3,...With the formula above, we are able to prove [1] interesting results about this largest real pole and its order.

There also exists a formula in terms of Newton polyhedra to calculate the topological zeta function (see J. Denef and F. Loeser, "Caractéristiques d'Euler-Poincaré, Fonctions Zêta Locales et Modifications Analytiques", J. Amer. Math. Soc. 5(4), pp. 705-720, 1992).

I developed a computer program (using Maple), based on both formulas to calculate Igusa's local zeta function for a fixed prime p and the topological zeta function. Of course this program only gives the chosen zeta function, if the polynomial is sufficiently non-degenerated. Nevertheless, this condition is "almost always" satisfied.

Last changes: 22-05-2002