In 2013, Giansiracusa and Giansiracusa have employed concepts from F1-geometry to endow the
set-theoretic tropicalization of a classical variety with the structure of a semiring scheme
over the tropical semifields T. The set-theoretic tropicalization can be retrieved from the
scheme-theoretic tropicalization by taking T-rational points. This discovery was the starting
point of tropical scheme theory, which is a recent new branch of tropical geometry.
Intuitively, the scheme-theoretic tropicalization can be seen as the base change of the
classical variety to T along a non-archimedean 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 set-theoretic tropicalization and the scheme-theoretic tropicalization in the sense of
Giansiracusa-Giansiracusa.
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