Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Tropical function fields, finite generation, and faithful tropicalization

Omid Amini, Shu Kawaguchi, JuAe Song

TL;DR

The paper provides a polyhedral, purely tropical characterization of the tropical function field associated with a strictly semistable pair $(\mathscr{X},\mathcal{H})$ and its Berkovich skeleton. It proves that every tropicalized rational function on $X_{\bar{K}}$ arises from a tropical function on the dual complex, establishing surjectivity and identifying the tropical function field with the tropicalization of $\operatorname{Rat}(X_{\bar{K}})$. It further shows that this tropical function field is finitely generated over the tropical rational semifield and furnishes a constructive approach to faithful tropicalization by explicitly producing coordinate functions whose tropicalizations realize a homeomorphism of the skeleton. The results extend Baker–Rabinoff’s one-dimensional picture to arbitrary dimensions, provide new proofs of finite generation in the discrete-valued setting, and yield explicit embeddings of skeleta into tropical spaces suitable for higher-dimensional applications.

Abstract

Given an algebraic variety defined over a discrete valuation field and a skeleton of its Berkovich analytification, the tropicalization process transforms function field of the variety to a semifield of tropical functions on the skeleton. Our main result offers a purely polyhedral characterization of this semifield: we show that a tropical function is in the image of the tropicalization map if and only if it takes the same slope near infinity along parallel half-lines of the skeleton. This extends a result of Baker and Rabinoff in dimension one to arbitrary dimensions. We use this characterization to establish that this semifield is finitely generated over the semifield of tropical rational numbers, providing a new proof of a recent result by Ducros, Hrushovski, Loeser and Ye in the discrete valued field case. As a second application, we present a new proof of the faithful tropicalization theorem by Gubler, Rabinoff and Werner in the discrete valuation field case. The proof is constructive and provides explicit coordinate functions for the embedding of the skeleton, extending the existing results in dimension one to arbitrary dimensions.

Tropical function fields, finite generation, and faithful tropicalization

TL;DR

The paper provides a polyhedral, purely tropical characterization of the tropical function field associated with a strictly semistable pair and its Berkovich skeleton. It proves that every tropicalized rational function on arises from a tropical function on the dual complex, establishing surjectivity and identifying the tropical function field with the tropicalization of . It further shows that this tropical function field is finitely generated over the tropical rational semifield and furnishes a constructive approach to faithful tropicalization by explicitly producing coordinate functions whose tropicalizations realize a homeomorphism of the skeleton. The results extend Baker–Rabinoff’s one-dimensional picture to arbitrary dimensions, provide new proofs of finite generation in the discrete-valued setting, and yield explicit embeddings of skeleta into tropical spaces suitable for higher-dimensional applications.

Abstract

Given an algebraic variety defined over a discrete valuation field and a skeleton of its Berkovich analytification, the tropicalization process transforms function field of the variety to a semifield of tropical functions on the skeleton. Our main result offers a purely polyhedral characterization of this semifield: we show that a tropical function is in the image of the tropicalization map if and only if it takes the same slope near infinity along parallel half-lines of the skeleton. This extends a result of Baker and Rabinoff in dimension one to arbitrary dimensions. We use this characterization to establish that this semifield is finitely generated over the semifield of tropical rational numbers, providing a new proof of a recent result by Ducros, Hrushovski, Loeser and Ye in the discrete valued field case. As a second application, we present a new proof of the faithful tropicalization theorem by Gubler, Rabinoff and Werner in the discrete valuation field case. The proof is constructive and provides explicit coordinate functions for the embedding of the skeleton, extending the existing results in dimension one to arbitrary dimensions.

Paper Structure

This paper contains 19 sections, 21 theorems, 86 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Polyhedral complexes
  3. Polyhedral complexes in $\mathbb{R} {\sp{}} !^n$
  4. Abstract polyhedral complex with a recession fan
  5. Tropical preliminaries
  6. Notation and conventions
  7. Semistable models and skeleta
  8. Tropicalization and the tropical function field of $({\mathscr{X}}, \mathcal{H})$
  9. Faithful tropicalization
  10. Semifields and finite generation
  11. Congruences
  12. Characterization theorem
  13. Construction of $f_\tau$
  14. Construction of $g_\tau$
  15. Proof of Theorem \ref{['thm:Q:1']}
  16. ...and 4 more sections

Key Result

Theorem 1.1

For every $f\in \operatorname{Rat}(X_{\bar{K}})$, the tropicalization $\operatorname{trop}(f)$ belongs to $\operatorname{Rat} (\hbox{[}1.1]{$$}\Delta({\mathscr{X}}, \mathcal{H})\hbox{[}1.1]{$$})$. The resulting map is surjective. Consequently, the tropical function field coincides with the semifield of tropicalized rational functions on $X_{\bar{K}}$.

Figures (2)

  • Figure 1: Barycentric subdivision of an extended simplex of dimension two with two vertices and a single ray.
  • Figure 2: Barycentric subdivision of an extended simplex with three vertices and a single ray. Only a portion of the subdivision is given, with four solids depicted in color: three tetrahedra with vertices $\{v_1, p \sb{\space\scaleobj{.8}{13}} , p \sb{\space\scaleobj{.8}{123}} , p \sb{\space\scaleobj{.8}{123\rho}} \}$, $\{v_1, p \sb{\space\scaleobj{.8}{13}} , p \sb{\space\scaleobj{.8}{13\rho}} , p \sb{\space\scaleobj{.8}{123\rho}} \}$, and $\{v_1, p \sb{\space\scaleobj{.8}{1\rho}} , p \sb{\space\scaleobj{.8}{13\rho}} , p \sb{\space\scaleobj{.8}{123\rho}} \}$, and a prism with vertices $\{ p \sb{\space\scaleobj{.8}{1\rho}} , p \sb{\space\scaleobj{.8}{13\rho}} , p \sb{\space\scaleobj{.8}{123\rho}} \}$ and a ray $\rho$.

Theorems & Definitions (47)

  • Theorem 1.1: Characterization Theorem
  • Theorem 1.2: Finite generation of tropical function fields
  • Theorem 1.3: Gubler--Rabinoff--Werner GRW16
  • Theorem 1.4
  • Definition 2.1: Facewise (integral $\mathbb{Q} {\sp{}} !$-)affine functions and $\operatorname{Rat}(\Sigma)$
  • Definition 2.2: Piecewise (integral $\mathbb{Q} {\sp{}} !$-)affine functions and $\operatorname{Rat} (\hbox{[}1.1]{$$}\Sigma\hbox{[}1.1]{$$})$
  • Definition 2.3: Convex function
  • Definition 2.4: Convexity of a polyhedral complex
  • Example 2.5
  • Proposition 3.1
  • ...and 37 more