Diophantine inequalities with power sums
Journal de Théorie des Nombres de Bordeaux, Volume 19 (2007) no. 2, pp. 547-560.

The ring of power sums is formed by complex functions on of the form

α(n)=b1c1n+b2c2n+...+bhchn,

for some b i ¯ and c i . Let F(x,y) ¯[x,y] be absolutely irreducible, monic and of degree at least 2 in y. We consider Diophantine inequalities of the form

|F(α(n),y)|<|Fy(α(n),y)|·|α(n)|-ε

and show that all the solutions (n,y)× have y parametrized by some power sums in a finite set. As a consequence, we prove that the equation

F(α(n),y)=f(n),

with f[x] not constant, F monic in y and α not constant, has only finitely many solutions.

On appelle somme de puissances toute suite α: de nombres complexes de la forme

α(n)=b1c1n+b2c2n+...+bhchn,

où les b i ¯ et les c i sont fixés. Soit F(x,y) ¯[x,y] un polynôme unitaire, absolument irréductible, de degré au moins 2 en y. On démontre que les solutions (n,y)× de l’inégalité

|F(α(n),y)|<|Fy(α(n),y)|·|α(n)|-ε

sont paramétrées par un nombre fini de sommes de puissances. Par conséquent, on déduit la finitude des solutions de l’équation diophantienne

F(α(n),y)=f(n),

f[x] est un polynôme non constant et α est une somme de puissances non constante.

Received:
Published online:
DOI: 10.5802/jtnb.601
Amedeo Scremin 1

1 Institut für Mathematik A Technische Universität Graz Steyrergasse 30 A-8010 Graz, Austria
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Amedeo Scremin. Diophantine inequalities with power sums. Journal de Théorie des Nombres de Bordeaux, Volume 19 (2007) no. 2, pp. 547-560. doi : 10.5802/jtnb.601. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.601/

[1] A. Baker, A sharpening of the bounds for linear forms in logarithms II. Acta Arithmetica 24 (1973), 33–36. | MR: 376549 | Zbl: 0261.10025

[2] P. Corvaja and U. Zannier, Diophantine Equations with Power Sums and Universal Hilbert Sets. Indag. Math., N.S. (3) 9 (1998), 317–332. | MR: 1692189 | Zbl: 0923.11103

[3] P. Corvaja and U. Zannier, Some new applications of the Subspace Theorem. Compositio Math. 131 (2002), 319–340. | MR: 1905026 | Zbl: 1010.11038

[4] R. Dvornicich and U. Zannier, On polynomials taking small values at integral arguments. Acta Arithmetica (2) 42 (1983), 189–196. | MR: 719248 | Zbl: 0515.10049

[5] M. Eichler, Introduction to the theory of algebraic numbers and functions. Academic press, New York and London. (1966). | MR: 209258 | Zbl: 0152.19502

[6] J.-H Evertse, An improvement of the quantitative subspace theorem. Compositio Math. (3) 101 (1996), 225–311. | Numdam | MR: 1394517 | Zbl: 0856.11030

[7] C. Fuchs, Exponential-Polynomial Equations and Linear Recurrences. PhD. thesis. Technische Universität Graz (2002).

[8] K. Iwasawa, Algebraic functions. Translations of Mathematical Monographs Vol. 118. American Mathematical Society, Providence, Rhode Island (1993). | MR: 1213914 | Zbl: 0790.14017

[9] P. Kiss, Differences of the terms of linear recurrences. Studia Sci. Math. Hungar. (1-4) 20 (1985), 285–293. | MR: 886031 | Zbl: 0628.10008

[10] A. Pethö, Perfect powers in second order linear recurrences. J. Number Theory 15 (1982), 5–13. | MR: 666345 | Zbl: 0488.10009

[11] W. M. Schmidt, Diophantine Approximations and Diophantine Equations. Lecture Notes in Math. vol. 1467. Springer-Verlag (1991). | MR: 1176315 | Zbl: 0754.11020

[12] W. M. Schmidt, Diophantine Approximation. Lecture Notes in Math. vol. 785. Springer-Verlag. (1980). | MR: 568710 | Zbl: 0421.10019

[13] A. Scremin, Tesi di Laurea “Equazioni e Disequazioni Diofantee Esponenziali”. Università degli Studi di Udine (2001).

[14] T. N. Shorey and C. L. Stewart, Pure powers in Recurrence Sequences and Some Related Diophantine Equations. J. Number Theory 27 (1987), 324–352. | MR: 915504 | Zbl: 0624.10009

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