# Diophantine inequalities involving several power sums

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## Abstract.

Let Open image in new window denote the ring of power sums, i.e. complex functions of the form Open image in new window for some Open image in new window and *α*_{ i } ∈ *A*, where Open image in new window is a multiplicative semigroup. Moreover, let Open image in new window We consider Diophantine inequalities of the form Open image in new window where *α*>1 is a quantity depending on the dominant roots of the power sums appearing as coefficients in *F*(*n*,*y*), and show that all its solutions Open image in new window have *y* parametrized by some power sums from a finite set. This is a continuation of the work of Corvaja and Zannier [4–6] and of the authors [10, 18] on such problems.

## Keywords

Complex Function Multiplicative Semigroup Diophantine Inequality Dominant Root## Preview

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## Notes

### Acknowledgments.

The first author was supported by the Austrian Science Foundation FWF, grant S8307-MAT. The second author was supported by Istituto Nazionale di Alta Matematica “Francesco Severi”, grant for abroad Ph.D. Both authors are most grateful to an anonymous referee for careful reading of the text and for several useful remarks improving previous versions of the paper.

## References

- 1.Baker, A.: A sharpening of the bounds for linear forms in logarithms II. Acta Arith.
**24**, 33–36 (1973)zbMATHGoogle Scholar - 2.Bugeaud, Y., Corvaja, P., Zannier, U.: An upper bound for the G.C.D. of
*a*^{n}-1 and*b*^{n}-1. Math. Zeitschrift**243**, 79–84 (2003)Google Scholar - 3.Cel, J.: The Newton-Puiseux theorem for several variables. Bull. Soc. Sci. Let. Łódź
**40**(11–20), 53–61 (1990)Google Scholar - 4.Corvaja, P., Zannier, U.: Diophantine equations with power sums and universal Hilbert sets. Indag. Math. New Ser.
**9**(3), 317–332 (1998)zbMATHGoogle Scholar - 5.Corvaja, P., Zannier, U.: On the diophantine equation
*f*(*a*^{m},*y*)=*b*^{n}. Acta Arith.**94**, 25–40 (2002)zbMATHGoogle Scholar - 6.Corvaja, P., Zannier, U.: Some new applications of the Subspace Theorem. Compos. Math.
**131**(3), 319–340 (2002)CrossRefzbMATHGoogle Scholar - 7.Evertse, J.-H.: An improvement of the Quantitative Subspace Theorem. Compos. Math.
**101**(3), 225–311 (1996)zbMATHGoogle Scholar - 8.Fuchs, C.: Exponential-polynomial equations and linear recurring sequences. Glas. Mat. Ser. III
**38**(58), 233–252 (2003)zbMATHGoogle Scholar - 9.Fuchs, C.: An upper bound for the G.C.D. of two linear recurring sequences. Math. Slovaca
**53**(1), 21–42 (2003)zbMATHGoogle Scholar - 10.Fuchs, C., Scremin, A.: Polynomial-exponential equations involving several linear recurrences. To appear in Publ. Math. Debrecen (Preprint:http://finanz.math. tu-graz.ac.at/~fuchs/eisr2.pdf)
- 11.Fuchs, C., Tichy, R.F.: Perfect powers in linear recurring sequences. Acta Arith.
**107.1**, 9–25 (2003)Google Scholar - 12.Krantz, S.G., Parks, H.R.: A Primer of Real Analytic Functions. Second Ed., Birkhäuser, Boston, 2002Google Scholar
- 13.Krantz, S.G., Parks, H.R.: The Implicit Function Theorem: History, Theory, and Applications. Birkhäuser, Boston, 2002Google Scholar
- 14.Pethő, A.: Diophantine properties of linear recursive sequences I. In: Bergum, G.E. (ed.) et al., Applications of Fibonacci numbers. Volume 7: Proceedings of the 7th international research conference on Fibonacci numbers and their applications, Graz, Austria, July 15–19, 1996, Kluwer Academic Publ., Dordrecht, 1998, pp. 295–309Google Scholar
- 15.Pethő, A.: Diophantine properties of linear recursive sequences II. Acta Math. Acad. Paed. Nyiregyháziensis
**17**, 81–96 (2001)MathSciNetGoogle Scholar - 16.Schmidt, W.M.: Diophantine Approximation. Springer Verlag, LN
**785**, 1980Google Scholar - 17.Schmidt, W.M.: Diophantine Approximations and Diophantine Equations. Springer Verlag, LN
**1467**, 1991Google Scholar - 18.Scremin, A.: Diophantine inequalities with power sums. J. Théor. Nombres Bordeaux, To appearGoogle Scholar
- 19.Shorey, T.N., Stewart, C.L.: Pure powers in recurrence sequences and some related Diophantine equations. J. Number Theory
**27**, 324–352 (1987)MathSciNetzbMATHGoogle Scholar