Author: Eiko

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

Introduction To Tropical Geometry

Tropical Numbers

The tropical semiring, depending on convention, is defined as

  • (R{},,) where is the maximum and is classical addition,

    xy=max(x,y)=xy,xy=x+y.

  • (R{+},,) where is the minimum and is classical addition,

    xy=min(x,y)=xy,xy=x+y.

We will follow the second convention.

Tropical Polynomials

The tropical monomials ax1e1x2e2xnen are understood as a classical function a+e1x1+e2x2++enxn. To distinguish tropical numbers and classical numbers, we suggest sometimes we use (a) to denote a tropical number aR{}, and make a convention that

  • If either an expression involves ,, or a bracket contained number (a), we understand the entire expression as tropical, e.g. (2)+(3)=(2) and 23=5.

A tropical polynomial is understood as a tropical sum of tropical monomials.

p(x1,,xn)=α(aα)xα

which evaluates as the function

p(x1,,xn)=minα(aα+xα):RnR.

which is

  • continuous, piecewise-linear with finite many pieces.

  • concave function

    p(λx+(1λ)y)λp(x)+(1λ)p(y),0λ1.

    (In the other convention it will be convex.)