A Continued Fractions Theory for the completion of the Puiseux field

TL;DR

This paper develops a continued fractions framework for the topological completion of the Puiseux field, establishing unique expansions, optimal rational approximations, and connections to Berkovich space dynamics.

Contribution

It introduces a novel continued fractions theory for Puiseux field completions, linking algebraic, Diophantine, and geometric aspects in a unified approach.

Findings

Unique continued fraction expansions for elements in the completion.

Best rational Diophantine approximations are achieved by these expansions.

Connections established between Berkovich space points and continued fractions.

Abstract

In this work, we study a continued fractions theory for the topological completion of the field of Puiseux series. As usual, we prove that any element in the completion can be developed as a unique continued fractions, whose coefficients are polynomials in roots of the variable, and that this approximation is the best ''rational'' Diophantine approximation of such element. Then, we interpret the preceding result in terms of the action of a suitable arithmetic subgroup of the special linear group on the Berkovich space defined over the said completion. We also explore the connections between points of type IV of the Berkovich space in terms of some ''non-convergent'' or ''undefined'' continued fractions, in a sense that we make precise in the text.