Equations in the Hadamard ring of rational functions
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 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 {a n } is a recurrence sequence and suppose that all the a n have a d th root in the field K; then (after possibly passing to a finite extension of K) one can choose a sequence of such d 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.
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, Série 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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