A quantitative version of Runge's theorem on diophantine equations

P. G. Walsh

Acta Arithmetica (1992)

  • Volume: 62, Issue: 2, page 157-172
  • ISSN: 0065-1036

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P. G. Walsh. "A quantitative version of Runge's theorem on diophantine equations." Acta Arithmetica 62.2 (1992): 157-172. <http://eudml.org/doc/206487>.

@article{P1992,
author = {P. G. Walsh},
journal = {Acta Arithmetica},
keywords = {polynomial diophantine equation; explicit upper bounds; size of the integer solutions; Runge's theorem},
language = {eng},
number = {2},
pages = {157-172},
title = {A quantitative version of Runge's theorem on diophantine equations},
url = {http://eudml.org/doc/206487},
volume = {62},
year = {1992},
}

TY - JOUR
AU - P. G. Walsh
TI - A quantitative version of Runge's theorem on diophantine equations
JO - Acta Arithmetica
PY - 1992
VL - 62
IS - 2
SP - 157
EP - 172
LA - eng
KW - polynomial diophantine equation; explicit upper bounds; size of the integer solutions; Runge's theorem
UR - http://eudml.org/doc/206487
ER -

References

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  1. [1] Y. André, G-functions and Geometry, Vieweg, Braunschweig 1989. 
  2. [2] M. Ayad, Sur le théorème de Runge, Acta Arith. 58 (1991), 203-209. Zbl0785.11018
  3. [3] J. Coates, Construction of rational functions on a curve, Proc. Cambridge Philos. Soc. 68 (1970), 105-123. Zbl0215.37302
  4. [4] B. M. Dwork and A. J. van der Poorten, The Eisenstein constant, Macquarie Math. reports, report No.90-0062R (1991). (To appear in Duke Math. J.). 
  5. [5] M. Eichler, Introduction to the Theory of Algebraic Numbers and Functions, Academic Press, London 1966. 
  6. [6] G. Eisenstein, Über eine allgemeine Eigenschaft der Reihen-Entwicklungen aller algebraischen Funktionen, Bericht Königl. Preuss. Akad. d. Wiss. zu Berlin (1852), 441-443. 
  7. [7] W. J. Ellison, Variations sur un thème de Carl Runge, Séminaire Delange-Pisot- Poitou 13 (1970-71), 9.01-9.04. 
  8. [8] A. Grytczuk and A. Schinzel, On Runge's theorem about diophantine equations, to appear in Colloq. Math. Soc. J. Bolyai 60, 1992. 
  9. [9] C. Hermite, Cours de M. Hermite rédigé en 1882, 4th ed., Hermann, Paris 1891. 
  10. [10] D. L. Hilliker and E. G. Straus, Determination of bounds for the solutions to those binary diophantine equations that satisfy the hypotheses of Runge's Theorem, Trans. Amer. Math. Soc. 280 (1983), 637-657. Zbl0528.10011
  11. [11] D. L. Hilliker and E. G. Straus, On Puiseux series whose curves pass through an infinity of algebraic lattice points, Bull. Amer. Math. Soc. (N.S.) 8 (1983), 59-62. Zbl0514.14009
  12. [12] D. W. Masser, Polynomial bound for Diophantine equations, Amer. Math. Monthly 93 (1980), 486-488. 
  13. [13] L. J. Mordell, Diophantine Equations, Academic Press, London 1969. 
  14. [14] C. Runge, Über ganzzahlige Lösungen von Gleichungen zwischen zwei Veränder- lichen, J. Reine Angew. Math. 100 (1887), 425-435. Zbl19.0076.03
  15. [15] A. Schinzel, An improvement of Runge's Theorem on diophantine equations, Comment. Pontificia Acad. Sci. 2 (1969), 1-9. Zbl0297.10009
  16. [16] W. M. Schmidt, Eisenstein's theorem on power series expansions of algebraic functions, Acta Arith. 56 (1990), 161-179. Zbl0659.12003
  17. [17] T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge University Press, Cambridge 1986. Zbl0606.10011
  18. [18] T. Skolem, Diophantische Gleichungen, J. Springer, Berlin 1938; reprinted by Chelsea, New York 1950. 
  19. [19] V. G. Sprindžuk, Classical Diophantine Equations in Two Unknowns, Nauka, Moskva 1982 (in Russian). 
  20. [20] J. Turk, On the difference between perfect powers, Acta Arith. 45 (1986), 289-307. Zbl0587.10010
  21. [21] R. J. Walker, Algebraic Curves, Princeton University Press, Princeton, New Jersey, 1950. Zbl0039.37701

Citations in EuDML Documents

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  1. P. G. Walsh, Corrections to "A quantitative version of Runge's theorem on diophantine equations" (Acta Arith. 62 (1992), 157-172)
  2. Rachid Boumahdi, Jesse Larone, Polynomials with values which are powers of integers
  3. Maohua Le, A note on the integer solutions ofhyperelliptic equations
  4. Aaron Levin, Variations on a theme of Runge: effective determination of integral points on certain varieties
  5. Dimitrios Poulakis, Estimation effective des points entiers d'une famille de courbes algébriques

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