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) Zbl 0696.10031 Bull. Soc. Math. Fr. 116, No. 4, 449-454 (1988). Let \({\mathbb{F}}\) 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 \({\mathbb{F}}\) which are transcendental over \({\mathbb{F}}(X)\). They consider a power series f over \({\mathbb{F}}((X))\), algebraic over \({\mathbb{F}}(X)\), and the p-adic numbers \(\lambda_ 1,...,\lambda_ s\). It is proved that \(f^{\lambda_ 1},...,f^{\lambda_ s}\) are algebraically independent over \({\mathbb{F}}(X)\) if and only if \(1,\lambda_ 1,...,\lambda_ s\) are linearly independent over \({\mathbb{Z}}\). They also obtain an extension of this result. A main subsidiary technique refers to the p-automatic sequences introduced by G. Christol, the second author, T. Kamae and G. Rauzy [Bull. Soc. Math. Fr. 108, 401-419 (1980; Zbl 0472.10035)]. Reviewer: D.Ştefănescu Cited in 1 ReviewCited in 3 Documents MSC: 11J85 Algebraic independence; Gel’fond’s method 11T99 Finite fields and commutative rings (number-theoretic aspects) 13F25 Formal power series rings 68Q70 Algebraic theory of languages and automata Keywords:algebraic independence; finite field; formal power series; p-automatic sequences Citations:Zbl 0599.12020; Zbl 0472.10035 PDFBibTeX XMLCite \textit{J. P. Allouche} et al., Bull. Soc. Math. Fr. 116, No. 4, 449--454 (1988; Zbl 0696.10031) Full Text: DOI Numdam EuDML References: [1] CHRISTOL (G.) , KAMAE (T.) , MENDÈS FRANCE (M.) et RAUZY (G.) . Suites algébriques, automates et substitutions , Bull. Soc. Math. France, t. 108, 1980 , p. 401-419. Numdam | MR 82e:10092 | Zbl 0472.10035 · Zbl 0472.10035 [2] GELFOND (A.O.) . Sur les nombres qui ont des propriétés additives et multiplicatives données , Acta Arith., t. 13, 1967 - 1968 , p. 259-265. Article | MR 36 #3745 | Zbl 0155.09003 · Zbl 0155.09003 [3] MENDÈS FRANCE (M.) et VAN DER POORTEN (A.J.) . Automata and the arithmetic of formal power series , Acta Arith., t. 46, 1986 , p. 211-214. Article | MR 88a:11064 | Zbl 0599.12020 · Zbl 0599.12020 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.