# Thomas’ conjecture over function fields

• [1] Institute of Analysis and Computational Number Theory Technische Universität Graz Steyrergasse 30, A-8010 Graz, Austria
• Volume: 19, Issue: 1, page 289-309
• ISSN: 1246-7405

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## Abstract

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Thomas’ conjecture is, given monic polynomials ${p}_{1},$$...,{p}_{d}\in ℤ\left[a\right]$ with $0<deg{p}_{1}<\cdots <deg{p}_{d}$, then the Thue equation (over the rational integers)$\left(X-{p}_{1}\left(a\right)Y\right)\cdots \left(X-{p}_{d}\left(a\right)Y\right)+{Y}^{d}=1$has only trivial solutions, provided $a\ge {a}_{0}$ with effective computable ${a}_{0}$. We consider a function field analogue of Thomas’ conjecture in case of degree $d=3$. Moreover we find a counterexample to Thomas’ conjecture for $d=3$.

## How to cite

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Ziegler, Volker. "Thomas’ conjecture over function fields." Journal de Théorie des Nombres de Bordeaux 19.1 (2007): 289-309. <http://eudml.org/doc/249951>.

@article{Ziegler2007,
abstract = {Thomas’ conjecture is, given monic polynomials $p_1,$$\ldots ,p_d \in \mathbb\{Z\}[a] with 0&lt;\deg p_1&lt; \cdots &lt;\deg p_d, then the Thue equation (over the rational integers)$(X-p\_1(a) Y) \cdots (X-p\_d(a) Y)+ Y^d=1$has only trivial solutions, provided a\ge a_0 with effective computable a_0. We consider a function field analogue of Thomas’ conjecture in case of degree d=3. Moreover we find a counterexample to Thomas’ conjecture for d=3.}, affiliation = {Institute of Analysis and Computational Number Theory Technische Universität Graz Steyrergasse 30, A-8010 Graz, Austria}, author = {Ziegler, Volker}, journal = {Journal de Théorie des Nombres de Bordeaux}, keywords = {Thue equation; function fields; cubic and quartic equations; ABC-theorem}, language = {eng}, number = {1}, pages = {289-309}, publisher = {Université Bordeaux 1}, title = {Thomas’ conjecture over function fields}, url = {http://eudml.org/doc/249951}, volume = {19}, year = {2007}, } TY - JOUR AU - Ziegler, Volker TI - Thomas’ conjecture over function fields JO - Journal de Théorie des Nombres de Bordeaux PY - 2007 PB - Université Bordeaux 1 VL - 19 IS - 1 SP - 289 EP - 309 AB - Thomas’ conjecture is, given monic polynomials p_1,$$\ldots ,p_d \in \mathbb{Z}[a]$ with $0&lt;\deg p_1&lt; \cdots &lt;\deg p_d$, then the Thue equation (over the rational integers)$(X-p_1(a) Y) \cdots (X-p_d(a) Y)+ Y^d=1$has only trivial solutions, provided $a\ge a_0$ with effective computable $a_0$. We consider a function field analogue of Thomas’ conjecture in case of degree $d=3$. Moreover we find a counterexample to Thomas’ conjecture for $d=3$.
LA - eng
KW - Thue equation; function fields; cubic and quartic equations; ABC-theorem
UR - http://eudml.org/doc/249951
ER -

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