Bounds for the solutions of unit equations

Bugeaud, Yann; Győry, Kálmán

doi:10.4064/aa-74-1-67-80

Cited by 67 publications

(143 citation statements)

References 14 publications

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“…, and R S is the S-regulator of K. For the S-regulator by using Lemma 3 of [11] and Lemma 2.1 of [2] (for the original result see Louboutin [21]) we can derive the bound…”

Section: ·1 the Results For The Number Field Casementioning

confidence: 99%

Effective results for unit points on curves over finitely generated domains

Bérczes

2015

Math. Proc. Camb. Phil. Soc.

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Let A be a commutative domain of characteristic 0 which is finitely generated over Z as a Z-algebra. Denote by A * the unit group of A and by K the algebraic closure of the quotient field K of A. We shall prove effective finiteness results for the elements of the setwhere F (X, Y ) is a non-constant polynomial with coefficients in A which is not divisible over K by any polynomial of the form X m Y n − α or X m − αY n , with m, n ∈ Z ≥0 , max(m, n) > 0, α ∈ K * . This result is a common generalization of effective results of Evertse and Győry (2013) on S-unit equations over finitely generated domains, of Bombieri and Gubler (2006) on the equation F (x, y) = 0 over S-units of number fields, and it is an effective version of Lang's general but ineffective theorem (1960) on this equation over finitely generated domains. The conditions that A is finitely generated and F is not divisible by any polynomial of the above type are essentially necessary.

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“…, and R S is the S-regulator of K. For the S-regulator by using Lemma 3 of [11] and Lemma 2.1 of [2] (for the original result see Louboutin [21]) we can derive the bound…”

Section: ·1 the Results For The Number Field Casementioning

confidence: 99%

Effective results for unit points on curves over finitely generated domains

Bérczes

2015

Math. Proc. Camb. Phil. Soc.

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“…By the properties of the Frobenius automorphism, α q ≡ β (mod q), so α q+1 ≡ βα ≡ −1 (mod q), therefore α 2(q+1) ≡ 1 (mod q) and a similar argument applies to β. This proves (5). Suppose now that q is odd.…”

Section: Introductionmentioning

confidence: 65%

“…This is Lemma 1 of [5] combined with a result of Voutier [33] (see [6]) giving a lower bound for the height of any non-zero algebraic number which is not a root of unity.…”

Section: A Bound For N In Terms Of Pmentioning

confidence: 99%

Almost powers in the Lucas sequence

Bugeaud¹,

Luca²,

Mignotte³

et al. 2008

J. Théor. Nombres Bordeaux

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A Henri Cohen,à l'occasion de son soixantième anniversaire Résumé. La liste complète des puissances pures qui apparaissent dans les suites de Fibonacci (F n ) n≥0 et de Lucas (L n ) n≥0 ne fut déterminée que tout récemment [10]. Les démonstrations combinent des techniques modulaires, issues de la preuve de Wiles du dernier théorème de Fermat, avec des méthodes classiques d'approximation diophantienne, dont la théorie de Baker. Dans le présent article, nous résolvons leséquations diophantiennes L n = q a y p , avec a > 0 et p ≥ 2, pour tous les nombres premiers q < 1087, et en fait pour tous les nombres premiers q < 10 6à l'exception de 13 d'entre eux. La stratégie suivie dans [10] s'avère inopérante en raison de la taille des bornes numériques données par les méthodes classiques et de la complexité deséquations de Thue qui apparaissent dans notreétude. La nouveauté mise en avant dans le présent article est l'utilisation simultanée de deux courbes de Frey afin d'aboutirà deséquations de Thue plus simples, et doncà de meilleures bornes numériques, qui contredisent les minorations que donne le crible modulaire.Abstract. The famous problem of determining all perfect powers in the Fibonacci sequence (F n ) n≥0 and in the Lucas sequence (L n ) n≥0 has recently been resolved [10]. The proofs of those results combine modular techniques from Wiles' proof of Fermat's Last Theorem with classical techniques from Baker's theory and Diophantine approximation. In this paper we solve the Diophantine equations L n = q a y p , with a > 0 and p ≥ 2, for all primes q < 1087, and indeed for all but 13 primes q < 10 6 . Here the strategy of [10] is not sufficient due to the sizes of the bounds and complicated nature of the Thue equations involved. The novelty in the present paper is the use of the double-Frey approach to simplify the Thue equations and to cope with the large bounds obtained from Baker's theory.

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“…Denote by s the cardinality of S and by R S the S-regulator of K (cf. [2]). For any algebraic number γ ∈ K of degree k the absolute height of γ is defined by…”

Section: Effective Upper Bounds For the Solutions Of Resultant Form Ementioning

confidence: 99%

“…We only give a sketch of the main steps of the standard arguments. For details consult [6] and [2]. (5) log |η…”

Section: An Efficient Algorithm For Solving Completely Resultant Formmentioning

confidence: 99%