Transcendence tests for Mahler functions

TL;DR

This paper introduces two novel transcendence tests for Mahler functions, one quick based on an eigenvalue and one universal based on Hankel matrices, applicable to functions of any degree.

Contribution

The paper presents the first transcendence tests for Mahler functions of arbitrary degree, utilizing eigenvalues and Hankel matrices for broad applicability.

Findings

Eigenvalue-based quick transcendence test for Mahler functions.

Universal transcendence test using Hankel matrices.

Applications demonstrated with multiple examples.

Abstract

We give two tests for transcendence of Mahler functions. For our first, we introduce the notion of the eigenvalue $\lambda_F$ of a Mahler function $F(z)$, and develop a quick test for the transcendence of $F(z)$ over $\mathbb{C}(z)$, which is determined by the value of the eigenvalue $\lambda_F$. While our first test is quick and applicable for a large class of functions, our second test, while a bit slower than our first, is universal; it depends on the rank of a certain Hankel matrix determined by the initial coefficients of $F(z)$. We note that these are the first transcendence tests for Mahler functions of arbitrary degree. Several examples and applications are given.