On the integer solutions of exponential equations in function fields

Umberto Zannier[1]

  • [1] Università degli studi di Udine, Dipartimento de Matematica e Informatica, Via delle scienze 206, 33100 Udine, (Italie)

Annales de l’institut Fourier (2004)

  • Volume: 54, Issue: 4, page 849-874
  • ISSN: 0373-0956

Abstract

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This paper is concerned with the estimation of the number of integer solutions to exponential equations in several variables, over function fields. We develop a method which sometimes allows to replace known exponential bounds with polynomial ones. More generally, we prove a counting result (Thm. 1) on the integer points where given exponential terms become linearly dependent over the constant field. Several applications are given to equations (Cor. 1) and to the estimation of the number of equal values of certain pairs of recurrence sequences (Cor. 2). In particular we substantially sharpen (Cor. 3) recent bounds for the number of integer solutions ( m , n ) of G m ( P ( X ) ) = c m , n G n ( X ) , where G n is a recurrence of polynomials, P is a polynomial and c m , n is a variable constant. Finally, we estimate the number of solutions to an S -unit type equation in two variables (Cor. 4), improving on known bounds.

How to cite

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Zannier, Umberto. "On the integer solutions of exponential equations in function fields." Annales de l’institut Fourier 54.4 (2004): 849-874. <http://eudml.org/doc/116135>.

@article{Zannier2004,
abstract = {This paper is concerned with the estimation of the number of integer solutions to exponential equations in several variables, over function fields. We develop a method which sometimes allows to replace known exponential bounds with polynomial ones. More generally, we prove a counting result (Thm. 1) on the integer points where given exponential terms become linearly dependent over the constant field. Several applications are given to equations (Cor. 1) and to the estimation of the number of equal values of certain pairs of recurrence sequences (Cor. 2). In particular we substantially sharpen (Cor. 3) recent bounds for the number of integer solutions $(m,n)$ of $G_m(P(X))=c_\{m,n\}G_n(X)$, where $G_n$ is a recurrence of polynomials, $P$ is a polynomial and $c_\{m,n\}$ is a variable constant. Finally, we estimate the number of solutions to an $S$-unit type equation in two variables (Cor. 4), improving on known bounds.},
affiliation = {Università degli studi di Udine, Dipartimento de Matematica e Informatica, Via delle scienze 206, 33100 Udine, (Italie)},
author = {Zannier, Umberto},
journal = {Annales de l’institut Fourier},
keywords = {number theory; diophantine equations; function fields; Diophantine equations; number of integer solutions; exponential equations in several variables; recurrence of polynomials; -unit type equation},
language = {eng},
number = {4},
pages = {849-874},
publisher = {Association des Annales de l'Institut Fourier},
title = {On the integer solutions of exponential equations in function fields},
url = {http://eudml.org/doc/116135},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Zannier, Umberto
TI - On the integer solutions of exponential equations in function fields
JO - Annales de l’institut Fourier
PY - 2004
PB - Association des Annales de l'Institut Fourier
VL - 54
IS - 4
SP - 849
EP - 874
AB - This paper is concerned with the estimation of the number of integer solutions to exponential equations in several variables, over function fields. We develop a method which sometimes allows to replace known exponential bounds with polynomial ones. More generally, we prove a counting result (Thm. 1) on the integer points where given exponential terms become linearly dependent over the constant field. Several applications are given to equations (Cor. 1) and to the estimation of the number of equal values of certain pairs of recurrence sequences (Cor. 2). In particular we substantially sharpen (Cor. 3) recent bounds for the number of integer solutions $(m,n)$ of $G_m(P(X))=c_{m,n}G_n(X)$, where $G_n$ is a recurrence of polynomials, $P$ is a polynomial and $c_{m,n}$ is a variable constant. Finally, we estimate the number of solutions to an $S$-unit type equation in two variables (Cor. 4), improving on known bounds.
LA - eng
KW - number theory; diophantine equations; function fields; Diophantine equations; number of integer solutions; exponential equations in several variables; recurrence of polynomials; -unit type equation
UR - http://eudml.org/doc/116135
ER -

References

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  6. V.I. Danilov, Algebraic Varieties and Schemes, Algebraic Geometry I 23 (1994), Springer-Verlag Zbl0787.14001
  7. J.-H. Evertse, On equations in two S -units over function fields of characteristic 0 , Acta Arith 47 (1986), 233-253 Zbl0632.10015MR870667
  8. J.-H. Evertse, K. Györy, On the number of solutions of weighted unit equations, Comp. Math 66 (1988), 329-354 Zbl0644.10015MR948309
  9. J.H. Evertse, H.P. Schlickewei, W.M. Schmidt, Linear equations in variables which lie in a multiplicative group, Annals of Math 155 (2002), 807-836 Zbl1026.11038MR1923966
  10. J.H. Evertse, U. Zannier, Linear equations with unknowns from a multiplicative group in a function field, (January 2004) Zbl1185.11022
  11. C. Fuchs, A. Pethö, R.F. Tichy, On the Diophantine Equation G n ( x ) = G m ( P ( x ) ) : Higher Order Recurrences Zbl1026.11040
  12. A. Schinzel, Polynomials with special regard to reducibility, vol. 77 (2000), Cambridge Univ. Press Zbl0956.12001MR1770638
  13. W.M. Schmidt, Linear Recurrence Sequences and Polynomial-Exponential Equations, Diophantine Approximation. Proc. of the C.I.M.E. Conference Cetraro (Italy, 2000) 1819 (2003), Springer-Verlag Zbl1034.11011
  14. U. Zannier, Some remarks on the S-unit equation in function fields, Acta Arith. LXIV (1993), 87-98 Zbl0786.11019MR1220487

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