Solution of Pisot’s conjecture on the Hadamard quotient of two rational functions. (Solution de la conjecture de Pisot sur le quotient de Hadamard de deux fractions rationnelles.) (French. Abridged English version) Zbl 0635.10007

van der Poorten, Alfred J.

C. R. Acad. Sci., Paris, Sér. I 306, No. 3, 97-102 (1988).

The author gives a sketch of a proof of Pisot’s conjecture on Hadamard quotients of rational functions. In a simple form the result states that if \(A(x)\) and \(B(x)\) are two power series expansions of rational functions with coefficients \(a_n\), \(b_n\in\mathbb Z\) such that \(b_n\) divides \(a_n\) for all \(n\), then the power series with coefficients \(a_n/b_n\) is an expansion of another rational function. The proper generalization to any ring, finitely generated over \(\mathbb Z\), is treated in this paper.

Reviewer: Frits Beukers (Utrecht)

MSC:

11B37

Recurrences

Keywords:

recurrence sequence; Pisot’s conjecture; Hadamard quotients of rational functions; power series expansions