Transcendence tests for Mahler functions

Abstract

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.

Computing solutions of linear Mahler equations

“…To this end, we briefly sketch Bell and Coons' "universal" transcendence test [6] and do a complexity analysis of their approach, using our notation and the same level of sophistication with regard to algorithms for subtasks. Define In this section, we drop the assumption 0 = 0.…”

“…Very recently, functional relations between Mahler functions have been further studied with a bias to effective tests and procedures [4,5,6,12,21]. Such studies motivate the need for algorithms that solve Mahler equations in various classes of functions.…”

“…of Bell and Coons [6] requires to compute truncations of Mahler series to suitable orders. So does the algorithm by Adamczewski and Faverjon [4,5] for the explicit computation of all linear dependence relations over Q between evaluations of Mahler functions at algebraic numbers.…”

“…Besides, Mahler functions being either rational or transcendental-but never algebraic-, solving Mahler equations for their rational functions is another natural approach to testing transcendence, and an alternative to Bell and Coons' (see further comments on this in §3. 6). Similarly, the hypertranscendence criterion by Dreyfus, Hardouin, and Roques [12] relies on determining if certain Mahler equations possess ramified rational solutions.…”

Becker’s conjecture on Mahler functions

“…There has been a flurry of recent activity involving the study of Mahler seriessee, e.g. [2,6,7,8,9,13,19,20, 21]-in large part due to the fact that one can often deduce transcendence of special values of Mahler series by knowing transcendence of the series itself, and also due to the guiding principle that much of the theory of Mahler series should mirror the much better developed theory of solutions to homogeneous differential equations.…”