An n-variable polynomial $p(x)=\bigoplus_{i=1}^n \beta_i \textbf{x}^{\alpha_i}$ (here $\textbf{x}$ is the tuple $(x_1,x_2,...,x_n)\in \mathbb{R}^n$) in the tropical (max-plus) semiring has an associated variety consisting of all points $y\in \mathbb{R}^n$ such that two or more monomials achieve their maximum on $y$, i.e. $\beta_j y^{\alpha_j}=\beta_i y^{\alpha_i}$ for at least two $i\neq j$. Clearly (infinitely) many polynomials may give rise to the same variety (one can add arbitrarily more monomials to the polynomial with very small coefficients, which will hence never be the maximum term in the polynomial). Hence given a set of $m$ polynomials, there are $k\leq m$ varieties that can be obtained from the set, where in some cases the inequality is strict.
I want to know more about this phenomenon, but I'm not sure what the terminology is. I want to say its something like "degeneracy" of polynomials, but I'm not sure. So my question is whether anyone knows how to count how many polynomials give rise to the same variety, or has a reference where this problem is discussed in detail.