Thomas’ conjecture over function fields

Volker Ziegler[1]

  • [1] Institute of Analysis and Computational Number Theory Technische Universität Graz Steyrergasse 30, A-8010 Graz, Austria

Journal de Théorie des Nombres de Bordeaux (2007)

  • Volume: 19, Issue: 1, page 289-309
  • ISSN: 1246-7405

Abstract

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Thomas’ conjecture is, given monic polynomials p 1 , ... , p d ∈ ℤ [ a ] with 0 < deg p 1 < ⋯ < deg p d , then the Thue equation (over the rational integers) ( X - p 1 ( a ) Y ) ⋯ ( X - p d ( a ) Y ) + Y d = 1 has only trivial solutions, provided a ≥ 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 -

References

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