Local Tropicalization
Patrick Popescu-Pampu, Dmitry Stepanov

TL;DR
This paper introduces a functorial framework for local tropicalization in commutative algebra, enabling the study of valuations and subvarieties in toroidal embeddings through polyhedral fans.
Contribution
It provides a general, functorial definition of local tropicalization that applies to valuations on rings and relates global and local tropicalizations in toric geometry.
Findings
Local tropicalizations are supports of finite rational polyhedral fans.
The framework generalizes tropicalization to local and toroidal settings.
Comparison between global and local tropicalizations is established.
Abstract
In this paper we propose a general functorial definition of the operation of \emph{local tropicalization} in commutative algebra. Let be a commutative ring, a finitely generated subsemigroup of a lattice, a morphism of semigroups, and the topological space of valuations on taking values in . Then we may \emph{tropicalize} with respect to any subset of the space of valuations . By definition, we get a subset of a rational polyhedral cone canonically associated to , enriched with strata at infinity. In particular, when is a local ring, is a \emph{local} morphism of semigroups, and is the space of valuations which are either positive or non-negative on , we call these processes \emph{local tropicalizations}. They depend only on the ambient toroidal structure,…
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Taxonomy
TopicsPolynomial and algebraic computation · Commutative Algebra and Its Applications · Logic, programming, and type systems
