Support of Laurent series algebraic over the field of formal power series

TL;DR

This paper investigates the support of algebraic Laurent series over formal power series fields, establishing a maximal dual cone, a gap theorem, and exploring diophantine implications in multiple variables.

Contribution

It introduces a maximal dual cone for the support of algebraic Laurent series and proves a gap theorem, advancing understanding of their structure and diophantine properties.

Findings

Existence of a maximal dual cone for the support.

A gap theorem for algebraic Laurent series.

Connections to diophantine properties of Laurent series fields.

Abstract

This work is devoted to the study of the support of a Laurent series in several variables which is algebraic over the ring of power series over a characteristic zero field. Our first result is the existence of a kind of maximal dual cone of the support of such a Laurent series. As an application of this result we provide a gap theorem for Laurent series which are algebraic over the field of formal power series. We also relate these results to diophantine properties of the fields of Laurent series.