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Let $f (x)$ be a power series $\sum_{n \geq 1} ζ (n) x^{e (n)}$ , where $(e (n))$ is a strictly increasing linear recurrence sequence of non-negative integers, and $(ζ (n))$ a sequence of roots of unity in ${\bar{ℚ}}_{p}$ satisfying an appropriate technical condition. Then we are mainly interested in characterizing the algebraic independence over $ℚ_{p}$ of the elements $f (α_{1}), ...,$ $f (α_{t})$ from $ℂ_{p}$ in terms of the distinct $α_{1}, ..., α_{t} \in ℚ_{p}$ satisfying $0 < | α_{τ} |_{p} < 1$ for $τ = 1, ..., t$ . A striking application of our basic result says that, in the case $e (n) = n$ , the set ${f (α) | α \in ℚ_{p}, 0 < | α |_{p} < 1}$ is algebraically independent over $ℚ_{p}$ if $(ζ (n))$ satisfies...

An inductive method for proving the transcendence of certain series

Daniel Duverney; Kumiko Nishioka — 2003

Acta Arithmetica

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Algebraic independence by Mahler’s method and $S$ -unit equations

Conditions for algebraic independence of certain power series of algebraic numbers

Algebraic Independence of Reciprocal Sums of Binary Recurrences.

New approach in Mahler's method.

Algebraic independence measures of the values of Mahler functions.

On an estimate for the orders of zeros of Mahler type functions

Algebraic independence of Fredholm series

Algebraic independence of elementary functions and its application to Masser's vanishing theorem.

Algebraic independence over $ℚ_{p}$

An inductive method for proving the transcendence of certain series