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.

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
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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/

[1] P. Corvaja, Rational fixed points for linear group actions, 2007, Preprint, available at http://www.arxiv.org/abs/math/0610661 to appear in Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5). | Numdam | MR | Zbl

[2] P. Corvaja and U. Zannier, Finiteness of integral values for the ratio of two linear recurrences, Invent. Math. 149 (2002), 431-451. | MR | Zbl

[3] P. Corvaja and U. Zannier, Some new applications of the Subspace Theorem, Compositio Math. 131 no. 3 (2002), 319-340. | MR | Zbl

[4] R. Dvornicich and U. Zannier, Cyclotomic diophantine problems (Hilbert irreducibility and invariant sets for polynomial maps), Duke Math. J. 139 (2007). | MR | Zbl

[5] A. Ferretti, Equazioni nell'anello di Hadamard, Master thesis, available at http: //etd.adm.unipi.it/theses/available/etd-08272004-153939/, 2004.

[6] C. Fuchs and A. Scremin, Diophantine inequalities involving several power sums, Manuscripta Math. 115 (2004), 163-178. | MR | Zbl

[7] C. Fuchs and A. Scremin, Polynomial-exponential equations involving several linear recurrences, Publ. Math. Debrecen 65 (2004), 149-172. | MR | Zbl

[8] J. H. Loxton, On the maximum modulus of cyclotomic integers, Acta Arith. 22 (1972), 69-85. | MR | Zbl

[9] Y. Pourchet, Solution du problème arithmétique du quotient de Hadamard de deux fractions rationnelles, C. R. Acad. Sci. Paris 288 (1979), 1055-1057. | MR | Zbl

[10] R. Rumely, Notes on van der Poorten's proof of the Hadamard quotient theorem, In: “Seminaire de Théorie des Nombres” Catherine Goldstein (ed.), Progress in Mathematics, no. 75, Birkhäuser, Boston - Basel, 1986-87, 349-409. | MR | Zbl

[11] R. Rumely and A. J. Van Der Poorten, A note on the Hadamard k th root of a rational function, J. Aust. Math. Soc. 43 (1987), 314-327. | MR | Zbl

[12] A. Schinzel, “Polynomials with Special Regard to Reducibility", Encyclopedia of mathematics and its applications, Vol. 77, Cambridge University Press, 2000. | MR | Zbl

[13] A. J. Van Der Poorten, Solution de la conjecture de Pisot sur le quotient de Hadamard de deux fractions rationnelles, C. R. Acad. Sci. Paris 306 (1988), 97-102. | MR | Zbl

[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. | MR | Zbl

[15] A. J. Van Der Poorten, A note on Hadamard roots of rational functions, Rocky Mountain J. Math. 26 (1996), 1183-1197. | MR | Zbl

[16] U. Zannier, A proof of Pisot’s d th root conjecture, Ann. of Math. 151 (2000), 375-383. | MR | Zbl

[17] U. Zannier, “Some Applications of Diophantine Approximation to Diophantine Equations", Forum Editrice, Udine, 2002.