Zeros of a Tropical polynomial

Question

Consider the Tropical semiring $(\mathbb{R}\cup\{-\infty\},\max,+)$. We define $x$ as a zero of a tropical polynomial $f(x)$ if $f$ attains its maximum twice at point $x$ in its linear parts. Why is this consistent with the intuitive idea that we are trying to solve $f(x)=-\infty$?

Answer 1

Here is an idea but I need to read some Universal Algebra to write it more formal. I remembered something that I heard in Category Theory course, the professor said and the following idea made me satisfied.

We use $f=0$ and $f-g$ for solving equations like $f=g$ but when subtraction and zero are not applicable or does not imply any special meaning we can go back to $f=g$ itself. So now in Tropical Algebraic Geometry that $f=-\infty$ is not giving us anything and we don't have subtraction too, we can consider a Tropical polynomial equation $\sum_{i=1}^nf_i=-\infty$ (which $f_i$s are terms of $f$) as union of answers of $\sum_{i\neq j,i=1..n}f_i=f_j$ for $j=1..n$. Which is exactly set of points that our polynomial admits its maximum in more than one of its terms.