# Christopher Hoyt

Harvey Mudd College Mathematics 2017

Thesis Proposal: On The Landscape of Random Tropical Curves Prof. Dagan Karp Prof. Mohamed Omar choyt@hmc.edu

## On The Landscape Of Random Tropical Curves

The tropical semiring is the set ℜ=ℝ∪{∞} that has the binary operators ⊕ and ⊗ (tropical addition and multiplication respectively) where for all x and y in ℜ, we have x⊕y=min(x,y) and x⊗y=x+y. When we exchange typical addition and multiplication with their tropical analogues, polynomials become convex, piecewise linear functions! In addition, tropical polynomials can have trivial terms because of the minimization operation of tropical addition. For example, one tropical polynomial p(x)=(0 ⊗ x ⊗ x) ⊕ (100 ⊗ x) ⊕ (0) = min{2x,100+x,0} has the the trivial term "(100 ⊗ x)."

Suppose that we generate random tropical curves where we fix the degree of the polynomial and the number of variables, but we draw all of the coefficients identically and independently from some random distribution. Then, what is the resulting distribution of the non-trivial terms in the polynomial? In this project, we will analyze random tropical curves from both an algebraic geometry approach as well as using Monte Carlo algorithms in order to determine the distribution of randomly generated tropical curves.