Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-29T01:23:21.757Z Has data issue: false hasContentIssue false
  • English
  • Français

Thue's equation over function fields

Published online by Cambridge University Press:  09 April 2009

Wolfgang M. Schmidt
Wolfgang M. Schmidt
Affiliation:
University of ColoradoBoulder, Colorado 80309, USA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Suppose we are given a “Thue equation” f(x, y) = 1, where f is a binary form with coefficients in a function field K of characteristic zero. A typical result is that if f is of degree at least 5 and has no multiple factors, then every solution x = (x, y) of the equation with components in K has H(x)≤90H(f) + 250g. Here g is the genus of K and H(x), H(f) are suitably defined heights. No assumption is made that x be “integral” in some sense. As an application, bounds are derived for “integral” solutions of hyperelliptic equations over K.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 25 , Issue 4 , June 1978 , pp. 385 - 422
Copyright
Copyright © Australian Mathematical Society 1978

References

Armitage, J. V. (1967), “Algebraic functions and an analogue of the geometry of numbers:The Riemann–Roch Theorem”, Arch. Math. 18, 383–393.CrossRefGoogle Scholar
Baker, A. (1968), “Contributions to the theory of diophantine equations (I), on the representation of integers by binary forms’, Phil. Trans. Royal Soc., London, A 263, 173–191.Google Scholar
Baker, A. (1969), “Bounds for the solutions of the hyperelliptic equation”, Proc. Camb. Phil. Soc. 65, 439–444.CrossRefGoogle Scholar
Davenport, H. (1965), “Onf3(t)–g2(t)rdquo;, Norske Vid. Selsk. Forh. (Trondheim), 38, 86–87.Google Scholar
Deuring, M. (1972), Lectures on the Theory of Algebraic Functions of One Variable (Springer Lecturer Notes 314)Google Scholar
Eichler, M. (1963), Einführung in die Theorie der algebraischen Zahlen und Funktionen (Birkhäuser Verlag, Basel and Stuttgart).CrossRefGoogle Scholar
Grauert, H. (1965), “Mordell's Vermutung über rationale Punkte auf algebraischen Kurven and Funktionenkörper”, Inst. Haut. Études 25, 131–149.CrossRefGoogle Scholar
Mahler, K. (1933a), “Zur Approximation algebraischer Zahlen (I). Über den grössten Primteiler binärer Formen”, Math. Ann. 107, 691–730.CrossRefGoogle Scholar
Mahler, K. (1933b), Über die rationalen Punkte auf Kurven vom Geschlecht Eins”, J. Reine Angew, Math. 170, 168–178.Google Scholar
Mahler, K. (1940), “An analogue to Minkowski's geometry of numbers in a field of series”, Ann. of Math. (2) 42, 305–320.Google Scholar
Ju., I. Manin (1963), “Rational points on algebraic curves over function fields”, Izv. Akad. Nauk Mat. 27, 1395–1440.Google Scholar
Osgood, C. F. (1973), “An effective lower bound on the ‘diophantine approximation’ of algebraic functions by rational functions”, Mathematika, 20, 4–15.CrossRefGoogle Scholar
Osgood, C. F. (1975), “Effective bounds on the ‘diophantine approximation’ of algebraic functions over fields of arbitrary characteristic and applications to differential equations”, Indag Math. 37, 105–119.CrossRefGoogle Scholar
Samuel, P. (1966), Lectures on Old and New Results on Algebraic Curves, Tata Inst., 36.Google Scholar
Schmidt, W. M. (1976), “On Osgood's effective Thue theorem for algebraic functions”, Commun. on Pure and Applied Math. 29, 759–773.CrossRefGoogle Scholar
Siegel, C. L. (1929), “Über einige Anwendungen diophantischer Approximationen”, Abh. d. Preuss. Akad. d. Wiss., Math. Phys. Kl. 1.Google Scholar
Thue, A. (1909), “Über Annäherungswerte algebraischer Zahlen”, Journal f. Math. 135, 284–305.Google Scholar
Uchiyama, S. (1961), “Rational approximations to algebraic functions”, J. Fac. Sci. Hokkaido Univ. Ser. 1, 15, 173–192.Google Scholar