REU 1995: P-adic 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 $f(x,y)=x^3-xy+y^3$
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 1988-present.
Go to Robinson's home page.