# Equations in the Hadamard ring of rational functions

Andrea Ferretti; Umberto Zannier

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2007)

- Volume: 6, Issue: 3, page 457-475
- ISSN: 0391-173X

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topFerretti, Andrea, and Zannier, Umberto. "Equations in the Hadamard ring of rational functions." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 6.3 (2007): 457-475. <http://eudml.org/doc/272259>.

@article{Ferretti2007,

abstract = {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 $\lbrace a_n\rbrace $ is a recurrence sequence and suppose that all the $a_n$ have a $d^\{\rm th\}$ root in the field $K$; then (after possibly passing to a finite extension of $K$) one can choose a sequence of such $d^\{\rm th\}$ 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 theHadamard 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.},

author = {Ferretti, Andrea, Zannier, Umberto},

journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},

language = {eng},

number = {3},

pages = {457-475},

publisher = {Scuola Normale Superiore, Pisa},

title = {Equations in the Hadamard ring of rational functions},

url = {http://eudml.org/doc/272259},

volume = {6},

year = {2007},

}

TY - JOUR

AU - Ferretti, Andrea

AU - Zannier, Umberto

TI - Equations in the Hadamard ring of rational functions

JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

PY - 2007

PB - Scuola Normale Superiore, Pisa

VL - 6

IS - 3

SP - 457

EP - 475

AB - 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 $\lbrace a_n\rbrace $ is a recurrence sequence and suppose that all the $a_n$ have a $d^{\rm th}$ root in the field $K$; then (after possibly passing to a finite extension of $K$) one can choose a sequence of such $d^{\rm th}$ 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 theHadamard 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.

LA - eng

UR - http://eudml.org/doc/272259

ER -

## References

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