Essentials of Tropical Combinatorics – Monograph

The tropical semi-ring is the triplet (R,⊕, ) with x⊕ y := min(x,y) and x y := x+ y . Its algebraic properties are somewhat modest, if one compares it with rings and fields, which are more common in most areas of mathematics. Still the distributive laws hold, and as an additional catch the addition is idempotent, that is, x⊕ x = x holds for all x ∈ R. As defined our tropical semi-ring is lacking an additive neutral element, but it is easy to fix that by adjoining ∞. In fact, this is what we will do frequently. Another immediate observation is that because of min(−x,−y) = −max(x,y) the semi-rings (R,min,+) and (R,max,+) are isomorphic. So it is just a matter of taste which one prefers over the other. However, we will encounter the situation where it is convenient to consider both additive structures at the same time. For now we stick to min for our tropical addition. This can be extended to vectors of real numbers by taking the minimum component-wise. This way we obtain a map ⊕ : Rd×Rd → Rd . Together with the tropical scalar multiplication : R×Rd → Rd : (λ ,x) 7→  λ x1 λ x2 .. λ xd =  λ+x1 λ+x2 .. λ+xd  we have the tropical semi-module (Rd ,⊕, ). An elementary approach to tropical geometry is to study multivariate polynomials and their evaluation maps in the tropical semi-module. This is the topic of this lecture.