Calculating Algebraic Invariants for Binary Forms of Degree 9 and 10

Jeff Hatley,  Mathematics (on left in photo)

Faculty Mentor: Dr. Thomas Hagedorn

              Invariant theory is an important area of mathematics that has been studied since the beginning of 20th century by such giants as Hilbert, Gordan, Young, and Sylvester. Despite receiving so much attention, however, many questions in invariant theory remain unanswered. In some cases, published “answers” have been found to be fallacious. It has been our hope over this summer to correct these mistakes and finally put to rest some problems in invariant theory.

              Our research has focused on the algebraic invariants of binary forms. A binary form is a degree n homogenous polynomial in two variables, and an algebraic invariant of such a form is a function of the coefficients whose value remains unchanged after the form undergoes a linear transformation. The simplest example would be the familiar discriminant of a quadratic polynomial which one learns in high school algebra class. For a quadratic polynomial of the form f(x,y)=ax2+bxy+cy2 , the discriminant is given by

b2-4ac, and it turns out to be an invariant.

              It has been our goal over this summer to accurately classify the minimal generating sets of invariants for binary forms of degree 9 and 10, a task which has been greatly facilitated by the invention of computer algebra systems, but which is still very difficult. We studied the many ways mathematicians have devised to represent and work with invariants, as well as creating a format of our own. Specifically, we sought to understand a classical method of working with invariants called the symbolic method. In the course of doing so, we also came up with our own way of representing and working with invariants, a form which we call H-Matrices.

We have implemented the computer algebra system Mathematica to aid in many calculations, and we have successfully verified previously published results on binary forms of degree 8 and below. Due to the large amounts of memory our current computer programs use, we have been unable to solve the degree 9 and 10 cases yet. That work is ongoing, as we continually refine our programs and search for faster methods of computing the invariants.

Personal Statement

              The SURP has been an enlightening and intellectually exciting way to spend the summer. It has allowed me to gain experience doing research in mathematics, and it has thus allowed me to become acquainted with the joys and frustrations of doing original work in mathematics. Working with my mentor, Dr. Thomas Hagedorn, and my peer, Glen Wilson, has allowed me to work on mathematics as part of a team, and I have learned so much about thinking and doing mathematics from both of them.

              Additionally, the weekly lunches were a nice chance to talk to students doing research in other disciplines and to share ideas and stories of our successes and failures. The community atmosphere made it very nice to be on campus during the summer.

              SURP was an incredibly rewarding experience, one which I would recommend to any interested student.