Transcendence tests for Mahler functions

2015/11/24 by Jason P. Bell, Bell, Jason P., Michael Coons +1
Mathematics · #11J91 #30B30 #39A06 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11J91 #msc:30B30 #msc:39A06

paper · pdf · doi:10.48550/arxiv.1511.07530

9 pages

arxiv created 2015/11/24 · arxiv updated 2015/11/25

Abstract

We give two tests for transcendence of Mahler functions. For our first, we introduce the notion of the eigenvalue λ_F of a Mahler function F(z), and develop a quick test for the transcendence of F(z) over ℂ(z), which is determined by the value of the eigenvalue λ_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.

Related

• M 'ethode de Mahler, transcendance et relations lin 'eaires : aspects\n effectifs
• Transcendence measure of e^1/n
• Transcendence of multivariate Mahler functions and algebraic relations between their values
• A new transcendence measure for the values of the exponential function at algebraic arguments
• A height gap theorem for coefficients of Mahler functions