A unifying approach to tropicalization

Abstract

In this paper, we introduce ordered blueprints and ordered blue schemes, which serve as a common language for the different approaches to tropicalizations and which enhances tropical varieties with a schematic structure. As an abstract concept, we consider a tropicalization as a moduli problem about extensions of a given valuation v : k → T between ordered blueprints k and T . If T is idempotent, then we show that a generalization of the Giansiracusa bend relation leads to a representing object for the tropicalization, and that it has yet another interpretation in terms of a base change along v. We call such a representing object a scheme theoretic tropicalization.This theory recovers and improves other approaches to tropicalizations as we explain with care in the second part of this text.The Berkovich analytification and the Kajiwara-Payne tropicalization appear as rational point sets of a scheme theoretic tropicalization. The same holds true for its generalization by Foster and Ranganathan to higher rank valuations.The scheme theoretic Giansiracusa tropicalization can be recovered from the scheme theoretic tropicalizations in our sense. We obtain an improvement due to the resulting blueprint structure, which is sufficient to remember the Maclagan-Rincón weights.The Macpherson analytification has an interpretation in terms of a scheme theoretic tropicalization, and we give an alternative approach to Macpherson's construction of tropicalizations.The Thuillier analytification and the Ulirsch tropicalization are rational point sets of a scheme theoretic tropicalization. Our approach yields a generalization to any, possibly nontrivial, valuation v : k → T with idempotent T and enhances the tropicalization with a schematic structure.Contents commutes. We denote by Val v (B, S) the set of valuations w : B → S over v.

A general framework for tropical differential equations

“…Several algebraic foundations for tropical geometry have been developed, including hyperfields [Vir11, BB19, Lor19, Jun21] Lorscheid's blueprints [Lor12,Lor15], and idempotent semirings [GG16, GG14, GG18, MR20a, MR18, MR20b, JM18b, JM18a, BE17, Yag16, CGM20]. For the present work, we find that idempotent semirings provide the most convenient language for the development of our theory.…”

Tropical geometry over the tropical hyperfield

“…In this text, we aim for a simplified account of scheme theoretic tropicalization, in contrast to the broader context of [15]. The specific situation of this paper-namely, the restriction to affine ordered blue schemes and the fixed tropicalization base T-allows us to simplify the exposition of our results considerably.…”

“…While hyperfields and hyperrings are coming short of certain properties that are required for tropical geometry, such as free objects and tensor products, the more ample notion of ordered blueprints has proven to be a suitable tool for tropical scheme theory (cf. [4] and [15]). Therefore we will refrain from spelling out the axiomatic of hyperfields, but we rather consider T, along with other algebraic objects of interest, as ordered blueprints.…”

“…In 2013, the seminal paper [8] by Jeffrey and Noah Giansiracusa inaugurated tropical scheme theory, a new branch of tropical geometry that seeks for a scheme theoretic formulation of tropical geometry. Maclagan and Rincón ([18]) showed soon after that the weights of tropical varieties are encoded in the scheme structure, and the author ( [15]) put this theory on a more sophisticated footing using ordered blueprints. This…”

Higher balancing for locally tropically convex tropical varieties

“…To conclude, we see higher balancing as an indication for that there might be interesting information about the tropicalizations of classical varieties that has not been used so far. Our hope is that this additional information finds a satisfactory explanation in terms of tropical scheme theory, as developed in [3], [7] and [8]. In particular, we would like to propose the following question as a guiding problem for further developments in tropical scheme theory.…”