At MSRI (Berkeley, CA). (In order from left to right) Dr. J. Maurice Rojas, Megan Ly, Christian McRoberts, Malachi Alexander, Ashley De Luna, Dr. Federico Ardila

Mentor: Dr. J. Maurice Rojas, Megan Ly
Malachi Alexander, Ashley De Luna, Christian McRoberts

Project Title: Topology of Positive Zero Sets of Bivariate Pentanomials
Abstract: A fundamental problem in many applications is determining the real solutions of a polynomial. In recent years, the A-discriminant variety of Gelfand, Kapranov and Zelevinsky has proved to be a valuable tool. Its complement over the real numbers defines regions of coefficient space called chambers. We implement an algorithm in MATLAB that plots the A-discriminant curve for a family of bivariate pentanomials and automatically computes the topology of all positive zero sets for the family of bivariate pentanomials. These positive zero sets are constant for all bivariate pentanomials within a given chamber of the A-discriminant. This algorithm provides automated topology computation for high degree polynomials and will be useful to the algebraic geometry community.

As part of this research experience, I visited the Mathematical Sciences Research Institute in Berkeley, California for six weeks during the summer of 2017. During the visit, I worked under the mentorship of Dr. J. Maurice Rojas and Megan Ly with two other undergraduates, Ashley De Luna and Christian McRoberts, on constructing an algorithm that computes the topology of positive zero sets of polynomials with two variables and five terms.

In San Francisco, CA.

We gave an oral presentation on our results at the conclusion of the program. We will be giving a poster presentation at SACNAS 2017 and JMM 2018 in the upcoming months.

Topology of Positive Zero Sets of Bivariate Pentanomials Talk (MSRI)
Topology of Positive Zero Sets of Bivariate Pentanomials Video (MSRI)
Topology of Positive Zero Sets of Bivariate Pentanomials Poster (CSUMB)

Additional Resources:
MSRI-UP 2017 Website