REU 1995: P-adic analysis and the Igusa local zeta function
(M. Robinson)

• 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.