Notes on van der Poorten’s proof of the Hadamard quotient theorem. I, II. (English) Zbl 0661.10017

Sémin. Théor. Nombres, Paris 1986-87, Prog. Math. 75, 349-382, 383-409 (1988).
[For the entire collection see Zbl 0653.00005.]
The aim of these papers is to give a complete and carefully detailed version of van der Poorten’s proof of the Hadamard quotient theorem: Let K be a field of characteristic 0. Let \(\sum_{n\geq 0}b_ nx^ n\) and \(\sum_{n\geq 0}c_ nx^ n\) be two formal power series in K[[x]], which are the expansions of rational functions, with \(b_ n\neq 0\) for n large enough. If all the quotients \(c_ n/b_ n\) belong to a finitely generated ring (over \({\mathbb{Z}})\), then the series \(\sum_{n\geq n_ 0}(c_ n/b_ n)x^ n\) is the expansion of a rational function.
Note that the condition on \((c_ n/b_ n)\) cannot be removed (take \(x/(1-x)=\sum_{n\geq 1}x^ n\), \(x/(1-x)^ 2=\sum_{n\geq 1}nx^ n\), although \(\sum_{n\geq 1}x^ n/n=-\log (1-x))\). The arguments of van der Poorten’s proof have been sketched first by Y. Pouchet [C. R. Acad. Sci., Paris, Sér. A 288, 1055-1057 (1979; Zbl 0421.13005)] and use crucially the Pólya-Cantor lemma.
In the papers under review the details of the proof are very carefully written down; the first paper deals with the case where the \(b_ n's\) and \(c_ n's\) belong to a fixed number field, in the second paper a specialization argument allows to consider the transcendental case.
Reviewer: J.-P.Allouche


11B37 Recurrences
13J05 Power series rings