Author: Eiko

Time: 2025-04-05 05:02:42 - 2025-04-05 05:02:42 (UTC)

References:

• Introduction to Tropical Geometry, Diane Maclagan, Bernd Sturmfels

Introduction To Tropical Geometry

Tropical Numbers

The tropical semiring, depending on convention, is defined as

• $(\mathbb{R}\cup \{-\mathrm{\infty }\},\oplus ,\odot )$ where $\oplus$ is the maximum and $\odot$ is classical addition,

  $$x\oplus y=max(x,y)=x\vee y,x\odot y=x+y.$$

• $(\mathbb{R}\cup \{+\mathrm{\infty }\},\oplus ,\odot )$ where $\oplus$ is the minimum and $\odot$ is classical addition,

  $$x\oplus y=min(x,y)=x\wedge y,x\odot y=x+y.$$

We will follow the second convention.

Tropical Polynomials

The tropical monomials $a\oplus x_1^{e_1}\odot x_2^{e_2}\odot \cdots \odot x_n^{e_n}$ are understood as a classical function $a+e_1{x}_1+e_2{x}_2+\cdots +e_n{x}_n$. To distinguish tropical numbers and classical numbers, we suggest sometimes we use $(a)$ to denote a tropical number $a\in \mathbb{R}\cup \{\mathrm{\infty }\}$, and make a convention that

• If either an expression involves $\oplus ,\odot$, or a bracket contained number $(a)$, we understand the entire expression as tropical, e.g. $(2)+(3)=(2)$ and $2\odot 3=5$.

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

$$p(x_1,\dots ,x_n)=\underset{\alpha }{min}(a_{\alpha }+x\cdot \alpha ):{\mathbb{R}}^n\to \mathbb{R}.$$

which is

• continuous, piecewise-linear with finite many pieces.

• concave function

  $$p(\lambda x+(1-\lambda )y)\ge \lambda \cdot p(x)+(1-\lambda )\cdot p(y),0\le \lambda \le 1.$$

  (In the other convention it will be convex.)

Polyhedral Geometry

Minkowski Sum

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 $Q$.

• $\text{NT}(fg)=\text{NT}(f)+\text{NT}(g)$.