Let be a number field. It is well known that the set of recurrencesequences with entries in 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 is a recurrence sequence and suppose that all the have a root in the field ; then (after possibly passing to a finite extension of ) one can choose a sequence of such 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 . 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.
@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}, pages = {457--475}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 6}, number = {3}, year = {2007}, mrnumber = {2370269}, zbl = {1150.11008}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2007_5_6_3_457_0/} }
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 SP - 457 EP - 475 VL - 6 IS - 3 PB - Scuola Normale Superiore, Pisa UR - http://www.numdam.org/item/ASNSP_2007_5_6_3_457_0/ LA - en ID - ASNSP_2007_5_6_3_457_0 ER -
%0 Journal Article %A Ferretti, Andrea %A Zannier, Umberto %T Equations in the Hadamard ring of rational functions %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2007 %P 457-475 %V 6 %N 3 %I Scuola Normale Superiore, Pisa %U http://www.numdam.org/item/ASNSP_2007_5_6_3_457_0/ %G en %F 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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