References:
The tropical semiring, depending on convention, is defined as
\((\mathbb{R}\cup \{-\infty\}, \oplus, \odot)\) where \(\oplus\) is the maximum and \(\odot\) is classical addition,
\[x\oplus y = \max(x,y) = x\vee y, \quad x\odot y = x + y.\]
\((\mathbb{R}\cup \{+\infty\}, \oplus, \odot)\) where \(\oplus\) is the minimum and \(\odot\) is classical addition,
\[x\oplus y = \min(x,y) = x\wedge y, \quad x\odot y = x + y.\]
We will follow the second convention.
The tropical monomials \(a\oplus x_1^{e_1}\odot x_2^{e_2}\odot \dots \odot x_n^{e_n}\) are understood as a classical function \(a+e_1x_1+e_2x_2+\cdots+e_nx_n\). To distinguish tropical numbers and classical numbers, we suggest sometimes we use \((a)\) to denote a tropical number \(a\in \mathbb{R}\cup\{\infty\}\), and make a convention that
A tropical polynomial is understood as a tropical sum of tropical monomials.
\[ p(x_1,\dots,x_n) = \sum_\alpha (a_\alpha) x^\alpha \]
which evaluates as the function
\[ p(x_1,\dots,x_n) = \min_\alpha (a_\alpha + x \cdot \alpha) :\mathbb{R}^n \to \mathbb{R}.\]
which is
continuous, piecewise-linear with finite many pieces.
concave function
\[p(\lambda x + (1-\lambda)y) \ge \lambda\cdot p(x) + (1-\lambda)\cdot p(y), \quad 0\le \lambda\le 1.\]
(In the other convention it will be convex.)
The Minkowski sum of two subsets \(A,B\subset \mathbb{R}^n\) is the sum of their components in \(\mathbb{R}^n\):
\[A+B = \{a+b \mid a\in A, b\in B\}.\]
\(\mathcal{N}_{P+Q} = \mathcal{N}_P \wedge \mathcal{N}_Q\), the normal fan of \(P+Q\) is the common refinement of the normal fans of \(P\) and \(Q\).
\(\NT(fg) = \NT(f) + \NT(g)\).