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

Abstract

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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 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.

How to cite

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Ferretti, 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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  14. [14] A. J. van der Poorten, Some facts that should be better known, especially about rational functions, In: “Number Theory and Applications”, Richard A. Mollin (ed.), Kluwer Academic Publishers, Dordrecht, 1989, 497–528. Zbl0687.10007MR1123092
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