Equations in the Hadamard ring of rational functions

Ferretti, Andrea; Zannier, Umberto

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 3, pp. 457-475.

Let $K$ be a number field. It is well known that the set of recurrencesequences with entries in $K$ is closed under component-wise operations, and so it can be equipped with a ring structure. We try to understand the structure of this ring, in particular to understand which algebraic equations have a solution in the ring. For the case of cyclic equations a conjecture due to Pisot states the following: assume $\left\{a_n\right\}$ is a recurrence sequence and suppose that all the $a_n$ have a $d^{\mathrm{th}}$ root in the field $K$; then (after possibly passing to a finite extension of $K$) one can choose a sequence of such $d^{\mathrm{th}}$ roots that satisfies a recurrence itself. This was proved true in a preceding paper of the second author. In this article we generalize this result to more general monic equations; the former case can be recovered for $g(X,Y)=X^d-Y=0$. Combining this with the Hadamard quotient theorem by Pourchet and Van der Poorten, we are able to get rid of the monic restriction, and have a theorem that generalizes both results.

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