Nathan Manwaring and Dr. Tyler Jarvis, Mathematics
Tropical Mathematics is a relatively new field of mathematics that is getting a lot of attention recently. It is applicable in the field of Combinatorics as well as Phylogenetics, and more uses are being found each day. Our goal was to find a way to classify Tropical Polynomials so that they may be better understood and ultimately bridge the gap between different applications and discoveries in Tropical Mathematics.
The first major discovery that we found was that there seemed to be a best way to express Tropical Polynomials. It is widely known that two different tropical polynomials can have the same graph and many have accepted that these kinds of polynomials are mathematically equivalent, but until our research no one has offered a standard way of expressing this relationship. We determined that there was indeed a best way of expressing these functions, which we have called “Least-coefficients form.” We also learned that by converting a polynomial to this form it is much easier to uncover its specific properties.
The discovery of least-coefficient polynomials brought on another interesting possibility, the “Fundamental Theory of Tropical Algebra.” Essentially, every single variable polynomial can be reduced to a unique set of irreducible factors. A proof of this theorem had been suggested by Zur Izhakian in 20051, but we created a much more simple and elegant proof using the properties of polynomials discovered by our standard expression technique. A form of this proof co-authored by Nathan Grigg and Nathan Manwaring is to be submitted for publication within the next few months.
One of the problems we encountered while trying to classify all tropical polynomials is the increased complexity as the number of variables increases. In one variable they are pretty straightforward. In two variables we can generally describe and classify them, but only for a few powers. And as expected it gets much more difficult to understand from that point on.
Although we were unable to make a classification of all two-variable polynomials we did discover another fundamental fact about tropical polynomials. We discovered a bridge between two areas of tropical mathematics. Combinatorists who use tropical algebra are likely to use balanced graphs which can all be generated by a tropical polynomial in two degrees. Algebraic Geometrists tend to focus on the actual polynomials themselves rather than the shapes they create. Our research group was able to determine a direct relationship between the polynomials and their dual graphs. That is given any two-variable polynomial we can determine the dual graph and vise versa. While this relationship is unfortunately as complex as the polynomials themselves, we can use this relationship to classify all of the conic polynomials. In layman terms, we simplified most of a master’s thesis to a few lines of a simple algorithm.
The discoveries made have been the topic of more than five presentations and so far one paper to be submitted for publication. More than that, our team of students has been able to research a new field of mathematics and make completely new discoveries in that field. In the future we plan to continue to search for a basic algorithm for classifying tropical polynomials. Using the discoveries already made as a springboard we hope to ultimately create an algorithm that will determine the set of ideals associated with any tropical polynomial. This would make possible more publications and continue to create a solid foundation of basic principles in tropical mathematics.