Advertisement

manuscripta mathematica

, Volume 115, Issue 2, pp 163–178 | Cite as

Diophantine inequalities involving several power sums

  • Clemens FuchsEmail author
  • Amedeo Scremin
Article

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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. 1.
    Baker, A.: A sharpening of the bounds for linear forms in logarithms II. Acta Arith. 24, 33–36 (1973)zbMATHGoogle Scholar
  2. 2.
    Bugeaud, Y., Corvaja, P., Zannier, U.: An upper bound for the G.C.D. of an-1 and bn-1. Math. Zeitschrift 243, 79–84 (2003)Google Scholar
  3. 3.
    Cel, J.: The Newton-Puiseux theorem for several variables. Bull. Soc. Sci. Let. Łódź 40 (11–20), 53–61 (1990)Google Scholar
  4. 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. 5.
    Corvaja, P., Zannier, U.: On the diophantine equation f(am,y)=bn. Acta Arith. 94, 25–40 (2002)zbMATHGoogle Scholar
  6. 6.
    Corvaja, P., Zannier, U.: Some new applications of the Subspace Theorem. Compos. Math. 131 (3), 319–340 (2002)CrossRefzbMATHGoogle Scholar
  7. 7.
    Evertse, J.-H.: An improvement of the Quantitative Subspace Theorem. Compos. Math. 101 (3), 225–311 (1996)zbMATHGoogle Scholar
  8. 8.
    Fuchs, C.: Exponential-polynomial equations and linear recurring sequences. Glas. Mat. Ser. III 38 (58), 233–252 (2003)zbMATHGoogle Scholar
  9. 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. 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. 11.
    Fuchs, C., Tichy, R.F.: Perfect powers in linear recurring sequences. Acta Arith. 107.1, 9–25 (2003)Google Scholar
  12. 12.
    Krantz, S.G., Parks, H.R.: A Primer of Real Analytic Functions. Second Ed., Birkhäuser, Boston, 2002Google Scholar
  13. 13.
    Krantz, S.G., Parks, H.R.: The Implicit Function Theorem: History, Theory, and Applications. Birkhäuser, Boston, 2002Google Scholar
  14. 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. 15.
    Pethő, A.: Diophantine properties of linear recursive sequences II. Acta Math. Acad. Paed. Nyiregyháziensis 17, 81–96 (2001)MathSciNetGoogle Scholar
  16. 16.
    Schmidt, W.M.: Diophantine Approximation. Springer Verlag, LN 785, 1980Google Scholar
  17. 17.
    Schmidt, W.M.: Diophantine Approximations and Diophantine Equations. Springer Verlag, LN 1467, 1991Google Scholar
  18. 18.
    Scremin, A.: Diophantine inequalities with power sums. J. Théor. Nombres Bordeaux, To appearGoogle Scholar
  19. 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

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Technische Universität GrazInstitut für MathematikGrazAustria
  2. 2.Technische Universität GrazInstitut für MathematikGrazAustria

Personalised recommendations