  For $i = 0, 1, \ldots, n$, let $C_i$ be independent and identically
distributed random variables with distribution $F$ with support $(0,\infty)$.
The number of zeros of the random tropical polynomials $\mathcal{T}f_n(x) =
\min_{i=1,\ldots,n}(C_i + ix)$ is also the number of faces of the lower convex
hull of the $n+1$ random points $(i,C_i)$ in $\mathbb{R}^2$. We show that this
number, $Z_n$, satisfies a central limit theorem when $F$ has polynomial decay
near $0$. Specifically, if $F$ near $0$ behaves like a $gamma(a,1)$
distribution for some $a > 0$, then $Z_n$ has the same asymptotics as the
number of renewals on the interval $[0,\log(n)/a]$ of a renewal process with
inter-arrival distribution $-\log(Beta(a,2))$. Our proof draws on connections
between random partitions, renewal theory and random polytopes. In particular,
we obtain generalizations and simple proofs of the central limit theorem for
the number of vertices of the convex hull of $n$ uniform random points in a
square. Our work leads to many open problems in stochastic tropical geometry,
the study of functionals and intersections of random tropical varieties.
