On the convergence of generalized power series solutions of $q$-difference equations

TL;DR

This paper establishes a sufficient condition for the convergence of generalized power series solutions to algebraic q-difference equations, based on geometric properties of the series' exponents, avoiding small divisors issues.

Contribution

It introduces a new geometric criterion for convergence of generalized power series solutions to q-difference equations, expanding understanding beyond previous algebraic approaches.

Findings

The convergence criterion depends on the semi-group of power exponents.

Examples illustrate cases where the criterion applies or fails.

The criterion avoids small divisors phenomena.

Abstract

A sufficient condition for the convergence of a generalized formal power series solution to an algebraic $q$-difference equation is provided. The main result leans on a geometric property related to the semi-group of (complex) power exponents of such a series. This property corresponds to the situation in which the small divisors phenomenon does not arise. Some examples illustrating the cases where the obtained sufficient condition can be or cannot be applied are also depicted.