Note sur un article de Sharif et Woodcock

2014 H. Sharif and C. Woodcock give in [26] a characterization of formal power series with coefficients in a field K of positive characteristic, which are algebraic over K(X). They deduce in a simple way from this theorem the algebraicity of Hadamard products and diagonals of algebraic power series. (These results have been also obtained by T. Harase [14]). We give here a slightly different proof of their theorem and we show how it can lead to an interesting generalization of the notion of pk-substitution on an infinite alphabet (included in a field of characteristic p). In the last part of this paper we come back to the algebraic independence of certain formal power series which have been previously studied in [2].

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