# REU 2005: Number Theory

(The picture above is of both REU groups and is thanks to Adrienne Rau. )
From left, Matthew Pragel, Rosica Dineva, Margaret Robinson, Jessica Sidman, Ricardo Portilla, Adiennne Rau, Jason Starr (Visitor), Frances Worek, Vince Lyzinski, Joanna Miles, Lisa Byrne, Sarah Gilles, Aaron Wolbach, Rex Cheung

## The student participants in the number theory group were:

- Rex Cheung (Brown University, '06), funded by NSF.

Local Derivatives and Bernstein Polynomials (paper)

- Rosica Dineva (Mount Holyoke College, '06), funded by MHC.

The Euler Totient, the Moebius, and the Divisor Functions (paper)

The Igusa Local Zeta Function (paper)

- Joanna Miles (Oberlin College,'06), funded by NSF.

Igusa Local zeta functions and their poles by the Newton polygon method (paper)

The mysterious actual order of the pole rho of the Igusa Local zeta function: Exploration of a conjecture (slides)

Zeros
Zeros is a program that finds the number of zeros and singular points mod p for a polynomial. We use it for finding N and S for the SPF or Newton polygon method of computing the Igusa Local Zeta Function. Options are provided for calculating N and S for polynomials in k variables, for either method of integration, and for a user-specified number of primes. This program was written by Joanna Miles and used by the 2005 Number Theory REU group to assist in hand calculations of the zeta function by allowing us to compare the number of zeros mod p.

- Ricardo Portilla (University of Virginia, '06), funded by NSF.

Local zeta functions of degenerate polynomials and poles associated with degeneracy (paper)

Degenerate Polynomials and poles: a look into degeneracy (slides)

- Matthew Pragel (University at Buffalo, '06), funded by NSF.

Techniques for computing the Igusa local zeta function (paper)

Undergraduate Research: Igusa local zeta function, poles, and me: a true life story (slides)

- Adrienne Rau (Barnard College, '06), funded by NSF.

Computing Igusa local zeta functions using a Newton polyhedron method for degenerate polynomials (paper)

Igusa local zeta functions and p-adic analysis: Newton polyhedra and degenerate polynomials (slides)

This work was produced under NSF grant number DMS-0353700. We
would also like to acknowledge the software package POLYGUSA that served as
important learning and experimental tool. This package was written by Kathleen Hoornaert and Davy Loots under the supervision of Jan Denef at Catholic University Leuven.

Polygusa

[REU home page
]
[ List of
REU projects since 1988 ] [ M. Robinson home page ]