Solutions of a Linear Equation in a Subgroup of Units in a Function Field

TL;DR

This paper proves that solutions to certain linear equations in the topological closure of a subgroup of units in a function field are actually solutions within the subgroup itself, advancing the understanding of function field analogs of classical conjectures.

Contribution

It establishes that for a broad class of function fields, solutions in the closure coincide with solutions in the subgroup, addressing specific cases of Skolem's conjecture in this context.

Findings

Solutions in the closure are contained in the subgroup

Addresses cases of Skolem's conjecture for function fields

Extends understanding of linear equations in algebraic structures

Abstract

Over a large class of function fields, we show that the solutions of some linear equations in the topological closure of a certain subgroup of the group of units in the function field are exactly the solutions that are already in the subgroup. This result solves some cases of the function field analog of an old conjecture proposed by Skolem.