In 2013, Giansiracusa and Giansiracusa have employed concepts from F1geometry to endow the
settheoretic tropicalization of a classical variety with the structure of a semiring scheme
over the tropical semifields T. The settheoretic tropicalization can be retrieved from the
schemetheoretic tropicalization by taking Trational points. This discovery was the starting
point of tropical scheme theory, which is a recent new branch of tropical geometry.
Intuitively, the schemetheoretic tropicalization can be seen as the base change of the
classical variety to T along a nonarchimedean valuation. To make this heuristics rigorous, we
need to pass from (semi)fields to hyperfields and from semiring schemes to blue schemes.
While these latter terminologies might appear exotic on a first sight, we will see that they
provide in fact a very simple and natural language for tropical scheme theory. In particular, we
can define the tropicalization literally as a base change along a valuation, which recovers
both, the settheoretic tropicalization and the schemetheoretic tropicalization in the sense of
GiansiracusaGiansiracusa.
