REU 1995: Padic analysis and the Igusa local zeta function
(M. Robinson)
The REU project for 1995 directed by Margaret Robinson investigated the
following
topics:
 The Igusa local zeta function for reducible curves.
This project continued the 1992 REU project
of Robinson.
 The Igusa local zeta function for Fermat hypersurfaces with exponent
$p^l$
The student participants were:
 Wungkum Fong, University of California, Berkeley '96
(currently a graduate student in mathematics at MIT)
 Sean Gray, Bucknell University '96 (currently a graduate student
in Meteorology at University of Maryland )
 Joel Grus, Rice University '96 (currently a graduate student in
mathematics at the University of Washington)
 Kristie Karlof, University of North Carolina at Chapel Hill '97
 Daniel Reuman, Harvard College '96 (currently a graduate student in
mathematics at University of Chicago)
 Rachel Stavenick, Mount Holyoke College '96 (currently a middle school
mathematics teacher)
The group produced the following papers, which are available in
postscript and gzipped postscript. (To unzip, execute the command "gunzip".)
 Joel Grus and Daniel Reuman, The Igusa local
zeta function for Fermat hypersurfaces with exponent $p^l$
 Abstract:
The purpose of this paper is to find a formula for the Igusa local zeta
function
for polynomials of type $x^{p^l}+y^{p^l}+z^{p^l}$. Two methods are examined,
the first of which is not very successful, and the second of which yields
a specific formula for the zeta function which applies
for large classes of $p$ and $l$. The second method also yields a specific
formula for the denominator of
the zeta function as well as a specific formula for the degree of the
numerator.
These apply for all $p$ and $l$.
 Sean Gray and Kristie Karlof,
The Stationary Phase formula for products of diagonal polynomials
 Abstract:
In this paper, we explore the calculation of the Igusa Local Zeta Function
for the product of two diagonal curves. We use Igusa's Stationary Phase
Formula (SPF) to first calculate a specific example. Then we generalize
to the product of any two diagonal curves.
 Rachel Stavenick,
Computing the Igusa local zeta function of $f(x,y)=x^3xy+y^3$
using resolution of singularities

Abstract:
This paper examines the method of resolution of singularities
used to compute Igusa local zeta functions. Specifically we look
at the resolution process as it applies to a specific example of the
local zeta function. We include a complete computation of the
example as well as a discussion of an alternate application of
the resolution process to our example.
Go to
List of REU projects 1988present.
Go to Robinson's home page.