POLYGUSA Version 0.5:
A computer program to calculate Igusa's local zeta function and
the topological zeta function for non-degenerated polynomials

Under supervision of Jan Denef and in collaboration with Davy Loots, mailto:davy.loots@siemens.atea.be, I developed a computer program to calculate Igusa's local zeta function associated to a polynomial f. This program works for polynomials which are sufficiently non-degenerated and is based on the formula in [1].

I also added some procedure to calculate the topological zeta function. This part of the program is based on the formula for the topological zeta function in [ 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].

One can also calculate the zeta function of the monodromy in the origin. This part of the program is based on the formula of [A.N. Varchenko, "Zeta-function of Monodromy and Newton's Diagram", Invent. Math. 37, pp. 253-262, 1976].

The program was written in Maple for Windows.

For the calculation of the Newton polyhedron we used the program "cdd". CDD stands for "double description code" and is written by Komei Fukuda.

Input:

• polynomial and its variables

Output:

• non-degenerated or degenerated
• Igusa's local zeta function (the global or the local one) or the topological zeta function (the global or the local one)
• all the candidate real poles of the chosen zeta function, their expected order and the responsible faces for the expected orders of those candidate poles
• - the points of the support of f
  - the vertices of the Newton polyhedron of f
  - the number of facets and proper faces
  - the defining system of inequalities of the Newton polyhedron
  - a picture of the Newton polyhedron, if the dimension is 2 or 3

• for each face t:
  - dimension
  - the points of the support of f_t
  - the directions of recession of t
  - the generators of D_t , the cone associated to t
  - a partition of D_t in simplicial cones
  - the multiplicities of the cones in the partition above.
  - for Igusa's local zeta function: the associated integer points, f_t , N_t , S_t and L_t
  - for the topological zeta function: J_t , Vol(t)*(dim t !)

• the Poincaré series of f, P(t)=S_{e in IN} p^-ne N_e t^e, and some numbers N_e = #{x in (Z_pe)ⁿ | f(x)=0 mod p^e}

• the zeta function of the monodromy in the origin

Technical issues

You need the following files:

• polygusa.mws (388 kB), the program written in Maple 5.1 or
  polyguv7.mws (406 kB), the program written in Maple 7.00
  You can also view the program-code and some examples in HTML-format, to look at it in your browser. More examples can be found in Appendix D of my Ph.D.
• cdd061.exe, a Windows 32bit-executable, that is called from within the Maple-file. You should adjust the file "polygusa.mws" to refer to the folder where you have downloaded the program.
  This executable was compiled from the distribution of Komei Fukuda (version 0.61) using MS Visual C++ 5. To compile your own version, you can download the source-code from his website.

The program has been tested on Windows 95/98 and 2000. Since Maple is available for other platforms, you could try to compile "cdd" for that platform to use the program. If someone converted it, they can always mail it so we can put it on the website.
Remark: when using Maple under Windows NT/2000 it is necessary to delete the file "opengl32.dll" in the folder "BIN.WNT" or the program can crash! This is a common bug in Maple.

Last changes: 7-6-2002