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Monatshefte für Mathematik

, Volume 180, Issue 2, pp 205–211 | Cite as

Elementary resolution of a family of quartic Thue equations over function fields

Article

Abstract

We consider and completely solve the parametrized family of Thue equations
$$\begin{aligned} X(X-Y)(X+Y)(X-\lambda Y)+Y^4=\xi , \end{aligned}$$
where the solutions xy come from the ring \(\mathbb {C}[T]\), the parameter \(\lambda \in \mathbb {C}[T]\) is some non-constant polynomial and \(0\ne \xi \in \mathbb {C}\). 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).

Keywords

Thue equation Families of Diophantine equations Function fields Determination of units 

Mathematics Subject Classification

11D25 

Notes

Acknowledgments

C. Fuchs and R. Paulin were supported by a grant of the Austrian Science Fund (FWF): P24574-N26, A. Jurasić was supported by the University of Rijeka research Grant No. 13.14.1.2.02 and by Croatian Science Foundation under the Project No. 6422. The authors are grateful to an anonymous referee for pointing out the geometric situation lying behind their line of reasoning.

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Copyright information

© Springer-Verlag Wien 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of SalzburgSalzburgAustria
  2. 2.Department of MathematicsUniversity of RijekaRijekaCroatia

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