Adelic amoebas disjoint from open halfspaces

TL;DR

This paper disproves a conjecture about adelic amoebas intersecting open halfspaces for certain subvarieties, and proposes a modified version supported by algebraic and tropical methods.

Contribution

It identifies limitations of the original conjecture and introduces a corrected version using algebraic, functorial, and torsion point techniques.

Findings

Counterexample for subvarieties of codimension > 1

Modified conjecture holds under new conditions

Application of Zhang's theorem on torsion points

Abstract

We show that a conjecture of Einsiedler, Kapranov, and Lind on adelic amoebas of subvarieties of tori and their intersections with open halfspaces of complementary dimension is false for subvarieties of codimension greater than one that have degenerate projections to smaller dimensional tori. We prove a suitably modified version of the conjecture using algebraic methods, functoriality of tropicalization, and a theorem of Zhang on torsion points in subvarieties of tori.