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Zbl 0696.10031
Allouche, J.-P.; Mendès-France, M.; van der Poorten, A.J.
Indépendance algébrique de certaines séries formelles. (Algebraic independence of certain formal power series). (French)
[J] Bull. Soc. Math. Fr. 116, No.4, 449-454 (1988). ISSN 0037-9484

Let ${\bbfF}$ be a finite field of characteristic p. The authors improve a result of the second and the third author [Acta Arith. 46, 211-214 (1986; Zbl 0599.12020)] about formal power series over ${\bbfF}$ which are transcendental over ${\bbfF}(X)$. They consider a power series f over ${\bbfF}((X))$, algebraic over ${\bbfF}(X)$, and the p-adic numbers $\lambda\sb 1,...,\lambda\sb s$. It is proved that $f\sp{\lambda\sb 1},...,f\sp{\lambda\sb s}$ are algebraically independent over ${\bbfF}(X)$ if and only if $1,\lambda\sb 1,...,\lambda\sb s$ are linearly independent over ${\bbfZ}$. They also obtain an extension of this result. A main subsidiary technique refers to the p-automatic sequences introduced by {\it G. Christol}, the second author, {\it T. Kamae} and {\it G. Rauzy} [Bull. Soc. Math. Fr. 108, 401-419 (1980; Zbl 0472.10035)].
[D.Ştefănescu]

MSC 2000:: *11J85 Algebraic independence results
11T99 Finite fields and commutative rings (number-theoretic aspects)
13F25 Formal power series rings
68Q70 Algebraic theory of automata

Keywords: algebraic independence; finite field; formal power series; p-automatic sequences

Citations: Zbl 0599.12020; Zbl 0472.10035

Cited in: Zbl 0725.68059

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