On linear independence measures of the values of Mahler functions

TL;DR

This paper develops methods to estimate how linearly independent the values of certain Mahler functions are, using Hermite-Padé approximations and functional equations to produce quantitative measures of independence.

Contribution

It introduces a novel approach combining Hermite-Padé approximations and functional equations to derive explicit linear independence measures for Mahler function values.

Findings

Established explicit linear independence measures for degree one and two Mahler functions.

Developed a technique based on determinants of Hermite-Padé approximation polynomials.

Provided a framework for future research on Mahler functions and their independence properties.

Abstract

In this paper, we estimate the linear independence measures for the values of a class Mahler functions of degree one and two. For the purpose, we study the determinants of suitable Hermite-Pad\'{e} approximation polynomials. Based on the non-vanishing of these determinants, we apply the functional equations to get an infinite sequence of approximations which is used to produce the linear independence measures.