Tropical analytic geometry, Newton polygons, and tropical intersections

TL;DR

This paper bridges tropical and rigid analytic geometry to derive formulas for valuations and multiplicities of zeros of convergent power series, and demonstrates that stable tropical intersections accurately compute algebraic multiplicities.

Contribution

It introduces new methods combining tropical and rigid analytic geometry to analyze Newton polygons and intersection multiplicities in non-Archimedean settings.

Findings

Provides a formula for valuations and multiplicities of common zeros

Shows stable tropical intersections compute algebraic multiplicities

Utilizes tropicalizations of rigid-analytic spaces

Abstract

In this paper we use the connections between tropical algebraic geometry and rigid analytic geometry in order to prove two main results. We use tropical methods to prove a theorem about the Newton polygon for convergent power series in several variables: if f_1,...,f_n are n convergent power series in n variables with coefficients in a non-Archimedean field K, we give a formula for the valuations and multiplicities of the common zeros of f_1,...,f_n. We use rigid-analytic methods to show that stable complete intersections of tropical hypersurfaces compute algebraic multiplicities even when the intersection is not tropically proper. These results are naturally formulated and proved using the theory of tropicalizations of rigid-analytic spaces, as introduced by Einsiedler-Kapranov-Lind [EKL06] and Gubler [Gub07b]. We have written this paper to be as readable as possible both to tropical and arithmetic geometers.