Graduation Year

2018

Document Type

Open Access Senior Thesis

Degree Name

Bachelor of Science

Department

Mathematics

Reader 1

Dagan Karp

Reader 2

Mohamed Omar

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Rights Information

(c) 2018 Christopher R Hoyt

Abstract

Tropical polynomials are similar to classical polynomials, however addition and multiplication are replaced with tropical addition (minimums) and tropical multiplication (addition). Within this new construction, polynomials become piecewise linear curves with interesting behavior. All tropical polynomials are piecewise linear curves, and each linear component uniquely corresponds to a particular monomial. In addition, certain monomial in the tropical polynomial can be trivial due to the fact that tropical addition is the minimum operator. Therefore, it makes sense to consider a graph of connectivity of the monomials for any given tropical polynomial. We investigate tropical polynomials where all coefficients are chosen from a standard normal distribution, and ask what the distribution will be for the graphs of connectivity amongst the monomials. We present a rudimentary algorithm for analytically determining the probability and show a Monte Carlo based confirmation for our results. In addition, we will give a variety of different theorems comparing relative likelihoods of different types of tropical polynomials.

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