Note on a paper of Sharif and Woodcock. (Note sur un article de Sharif et Woodcock.) (French) Zbl 0714.12006

The title concerns the article by H. Sharif and C. F. Woodcock, J. Lond. Math. Soc., II. Ser. 37, No. 3, 395–403 (1988; Zbl 0612.12018).
The aim of this paper is to review recent works about algebraic power series. More precisely, let \(k\) be a field of positive characteristic \(p\). It is shown that a power series in \(k[[x_ 1,...,x_ r]]\) is algebraic over \(k(x_ 1,...,x_ r)\) if and only if the sequence of its Taylor coefficients is generated by a \(p\)-substitution. As there are several variables and as the field is not finite, a generalisation of the notion of \(p^ k\)-substitution is needed. This generalisation, deeply connected with Schützenberger’s recognizability, is interesting by itself. Some applications are given, for instance diagonals of algebraic power series are proven to be algebraic, and many references are given.
The last part of the paper is devoted to prove algebraic independence over \(k(x_ 1,...,x_ r)\) of power series of the type \((1+f)^{\lambda}\) for algebraic \(f\) and \({\mathbb{Z}}\)-linearly independent \(\lambda\).
Reviewer: G.Christol


12E99 General field theory
11B85 Automata sequences
11J81 Transcendence (general theory)
11J85 Algebraic independence; Gel’fond’s method
13F25 Formal power series rings


Zbl 0612.12018
Full Text: DOI Numdam EuDML


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