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Archiv der Mathematik
Algebraic independence of certain Mahler functions
Masaaki Amou and Keijo V
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 ≥ 2andt 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) ∈Fof the form
,R(z) ∈R. (1.1)
It is seen that f (z) satisﬁes the following Mahler functional equation
Conversely, the functional equation (1.2) has a unique solution in F which is
given by the inﬁnite product (1.1).
There is a way to see when an element f(z) given by (1.1) belongs to
R.Thatis,iff(z) ∈R, then R(z)=f(z)/f(z
. Conversely, if R(z)=