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Indépendance algébrique de certaines séries formelles. (Algebraic independence of certain formal power series). (French) Zbl 0696.10031

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

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
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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
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