# Algebraic independence of certain Mahler functions

Algebraic independence of certain Mahler functions Arch. Math. 2018 Springer International Publishing AG, part of Springer Nature Archiv der Mathematik https://doi.org/10.1007/s00013-018-1196-7 ¨¨ ¨ Masaaki Amou and Keijo Vaananen Abstract. We prove algebraic independence of functions satisfying a sim- ple form of algebraic Mahler functional equations. The main result (The- orem 1.1) partly generalizes a result obtained by Kubota. This result is deduced from a quantitative version of it (Theorem 2.1), which is proved by using an inductive method originated by Duverney. As an applica- tion we can also generalize a recent result by Bundschuh and the second named author (Theorem 1.2 and its corollary). Mathematics Subject Classiﬁcation. Primary 11J85; Secondary 11J91. Keywords. Algebraic independence, Mahler function, Inﬁnite product. 1. Introduction. Throughout this paper let d ≥ 2and t denote positive inte- gers. Let K be an arbitrary ﬁeld of characteristic zero, and K[[z]] be the ring of formal power series in z with coeﬃcients from K. Set F := 1 + zK[[z]] and R := F∩ K(z). Note that F is a group under the usual multiplication on K[[z]] and R is a subgroup of F . We are interested in an arbitrary element f (z) ∈F of the form ν ν d t f http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archiv der Mathematik Springer Journals

# Algebraic independence of certain Mahler functions

, Volume OnlineFirst – May 29, 2018
11 pages

/lp/springer_journal/algebraic-independence-of-certain-mahler-functions-AjFGudngJO
Publisher
Springer International Publishing
Copyright © 2018 by Springer International Publishing AG, part of Springer Nature
Subject
Mathematics; Mathematics, general
ISSN
0003-889X
eISSN
1420-8938
D.O.I.
10.1007/s00013-018-1196-7
Publisher site
See Article on Publisher Site

### Abstract

Arch. Math. 2018 Springer International Publishing AG, part of Springer Nature Archiv der Mathematik https://doi.org/10.1007/s00013-018-1196-7 ¨¨ ¨ Masaaki Amou and Keijo Vaananen Abstract. We prove algebraic independence of functions satisfying a sim- ple form of algebraic Mahler functional equations. The main result (The- orem 1.1) partly generalizes a result obtained by Kubota. This result is deduced from a quantitative version of it (Theorem 2.1), which is proved by using an inductive method originated by Duverney. As an applica- tion we can also generalize a recent result by Bundschuh and the second named author (Theorem 1.2 and its corollary). Mathematics Subject Classiﬁcation. Primary 11J85; Secondary 11J91. Keywords. Algebraic independence, Mahler function, Inﬁnite product. 1. Introduction. Throughout this paper let d ≥ 2and t denote positive inte- gers. Let K be an arbitrary ﬁeld of characteristic zero, and K[[z]] be the ring of formal power series in z with coeﬃcients from K. Set F := 1 + zK[[z]] and R := F∩ K(z). Note that F is a group under the usual multiplication on K[[z]] and R is a subgroup of F . We are interested in an arbitrary element f (z) ∈F of the form ν ν d t f

### Journal

Archiv der MathematikSpringer Journals

Published: May 29, 2018

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