Degree of 1-dimensional tropical prevariety

Abstract

For a tropical prevariety $V\subset \RR^n$ (being a finite union of rational polyhedra) we define a tropical Hilbert function $T{H}_V(k)$ to be the maximal number of tropically independent on $V$ among tropical monomials with degrees at most $k$. In case $\dim V=1$ we define the tropical degree as $$degT(V):=\underset{k\to \infty }{\lim }\frac{T{H}_V(k)}{k}$$ \noindent and prove existence of the limit. We calculate explicitly (a modification of) the tropical degree of $V$ when $V={\cup }_l{V}_l$ is a star, i.e. the union of rays $V_l$ with a common apex.