Elementary resolution of a family of quartic Thue equations over function fields View Full Text


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Article Info

DATE

2016-01-09

AUTHORS

Clemens Fuchs, Ana Jurasić, Roland Paulin

ABSTRACT

We consider and completely solve the parametrized family of Thue equations X(X-Y)(X+Y)(X-λY)+Y4=ξ,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} X(X-Y)(X+Y)(X-\lambda Y)+Y^4=\xi , \end{aligned}$$\end{document}where the solutions x, y come from the ring C[T]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {C}[T]$$\end{document}, the parameter λ∈C[T]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \in \mathbb {C}[T]$$\end{document} is some non-constant polynomial and 0≠ξ∈C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\ne \xi \in \mathbb {C}$$\end{document}. It is a function field analogue of the family solved by Mignotte, Pethő and Roth in the integer case. A feature of our proof is that we avoid the use of height bounds by considering a smaller relevant ring for which we can determine the units more easily. Because of this, the proof is short and the arguments are very elementary (in particular compared to previous results on parametrized Thue equations over function fields). More... »

PAGES

205-211

References to SciGraph publications

  • 1994-09. A note on Thue's equation over function fields in MONATSHEFTE FÜR MATHEMATIK
  • 2005-11-16. On a Family of Thue Equations Over Function Fields in MONATSHEFTE FÜR MATHEMATIK
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    http://scigraph.springernature.com/pub.10.1007/s00605-015-0864-y

    DOI

    http://dx.doi.org/10.1007/s00605-015-0864-y

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