Equations in the Hadamard ring of rational functions
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 6 (2007) no. 3, p. 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 $g\left(X,Y\right)={X}^{d}-Y=0$. 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.

Classification:  11B37,  12E25,  13F25
@article{ASNSP_2007_5_6_3_457_0,
author = {Ferretti, Andrea and Zannier, Umberto},
title = {Equations in the Hadamard ring of rational functions},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 6},
number = {3},
year = {2007},
pages = {457-475},
zbl = {1150.11008},
mrnumber = {2370269},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2007_5_6_3_457_0}
}

Ferretti, Andrea; Zannier, Umberto. Equations in the Hadamard ring of rational functions. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 6 (2007) no. 3, pp. 457-475. http://www.numdam.org/item/ASNSP_2007_5_6_3_457_0/

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