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September 2011 Solving explicitly $F(x,y)=G(x,y)$ over function fields
István Gaál, Michael Pohst
Funct. Approx. Comment. Math. 45(1): 79-88 (September 2011). DOI: 10.7169/facm/1317045233


Consider binary forms $F(x,y), G(x,y)$ with coefficients in $\mathbb{Q}[t]$, assume that $F$ is irreducible. We give effective upper bounds for the heights of the solutions and an efficient algorithm to solve \[ w\cdot F(x,y)=z\cdot G(x,y) \] \[ in x,y\in \mathbb{Q}[t], w,z\in \mathbb{Q}[t]\cap U_S, \gcd(x,y)=1, \gcd(w,z)=1, \] where $U_S$ denotes a group of $S$-units in $\mathbb{Q}(t)$. We derive that there are only finitely many solutions up to constant factors. We also show that this is not true for global function fields. This is a generalization of the well known Thue equations. Effective upper bounds for the solutions of this general equation were given over number fields but it was not yet considered over function fields. We illustrate our method with a detailed numerical example.


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István Gaál. Michael Pohst. "Solving explicitly $F(x,y)=G(x,y)$ over function fields." Funct. Approx. Comment. Math. 45 (1) 79 - 88, September 2011.


Published: September 2011
First available in Project Euclid: 26 September 2011

zbMATH: 1127.11025
MathSciNet: MR2865414
Digital Object Identifier: 10.7169/facm/1317045233

Primary: 11Y50
Secondary: 11D57

Rights: Copyright © 2011 Adam Mickiewicz University


Vol.45 • No. 1 • September 2011
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