recently I've been reading about tropical polynomials and stumbled upon Andreas Gathmann's lecture notes.
In section 1.4, it is stated that a tropical polynomial in the form of
$$ g(x, y) = \max_i \{a_1^{(i)} x + a_2^{(i)} y \} $$
as a function depends on only the points $ (a_1^{(i)}, a_2^{(i)} )$ that are vertices of its Newton polygon $ \text{Newt}(g) = \text{conv}\{ (a_1^{(i)}, a_2^{(i)}) \} $
I can understand that this also generalizes to newton polytopes in higher dimensions, since all interior points will satisfy $ \mathbf{a}^T \mathbf{x} < \mathbf{a}_{k}^T \mathbf{x} $, where $\mathbf{a}_k$ is a vertex of the Newton Polytope.
What happens if $g$ also includes constant terms in its factors? E.g. if
$$ g(x, y) = \max_i \{ a_1^{(i)} x + a_2^{(i)} y + b_i \} $$
Can we incorporate in some way the constant terms in the Newton polytope construction to extend the above statement concerning the interior points?