Familles d'équations de Thue–Mahler n'ayant que des solutions triviales

Waldschmidt

2017

Mathematika

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Abstract. Let K be an algebraic number field of degree d 3, σ 1 , σ 2 , . . . , σ d the embeddings of K into C, α a non-zero element in K , a 0 ∈ Z, a 0 > 0 andLet υ be a unit in K . For a ∈ Z, we twist the binary formGiven m > 0, our main result is an effective upper bound for the size of solutions (x, y, a) ∈ Z 3 of the Diophantine inequalitiesfor which x y = 0 and Q(αυ a ) = K . Our estimate is explicit in terms of its dependence on m, the regulator of K and the heights of F 0 and of υ; it also involves an effectively computable constant depending only on d. §1. Introduction and the main results. Let d 3 be a given integer. We denote by κ 1 , κ 2 , . . . , κ 38 positive effectively computable constants which depend only on d. Let K be a number field of degree d. Denote by σ 1 , σ 2 , . . . , σ d the embeddings of K into C and by R the regulator of K . Let α ∈ K , α = 0, and let a 0 ∈ Z, a 0 > 0, be such that the coefficients of the polynomialare in Z. Let υ be a unit in K , not a root of unity. For a ∈ Z, define the polynomial f a (X ) in Z[X ] and the binary form

Section: Introduction and The Main Resultsmentioning

confidence: 99%

Families of Thue Equations Associated With a Rank One Subgroup of the Unit Group of a Number Field

Waldschmidt

2017

Mathematika

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“…The proof we gave in § 3 relies on our result on Diophantine equations in [3], which is a consequence of Schmidt's Subspace Theorem, while the proof of Corvaja and Zannier in [2] uses directly Schmidt's fundamental result on linear forms in algebraic numbers. It is likely that an improvement of our result could be achieved by adapting the arguments of [2] -so one would expect to obtain a refinement of our conclusion which would also include the statement of Theorem 5.1.…”

Section: Comparison With a Results Of Corvaja And Zanniermentioning

confidence: 99%

“…Theorem 3.1 is a corollary of the following theorem whose proof can be found in [3]. Proof (of Theorem 3.1).…”

Section: A Refinement Of Liouville's Estimatementioning

confidence: 99%

“…In a previous paper [3], we proved that certain families of diophantine equations have only trivial solutions. In this note (Theorem 3.1), we show how to deduce from our results on families of diophantine equations [3] some results on the approximation of an algebraic number by products of rational numbers and units.…”

Section: Introductionmentioning

confidence: 99%

“…In this note (Theorem 3.1), we show how to deduce from our results on families of diophantine equations [3] some results on the approximation of an algebraic number by products of rational numbers and units. Since the proofs rest on Schmidt's Subspace Theorem, these results are non-effective.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Approximation of an Algebraic Number by Products of Rational Numbers and Units

Waldschmidt²

2012

J. Aust. Math. Soc.

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Families of Cubic Thue Equations with Effective Bounds for the Solutions

Springer Proceedings in Mathematics &Amp; Statistics

Waldschmidt

2013

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To each non totally real cubic extension K of Q and to each generator α of the cubic field K, we attach a family of cubic Thue equations, indexed by the units of K, and we prove that this family of cubic Thue equations has only a finite number of integer solutions, by giving an effective upper bound for these solutions. StatementsLet us consider an irreductible binary cubic form having rational integers coefficientswith the property that the polynomial F(X, 1) has exactly one real root α and two complex imaginary roots, namely α ′ and α ′ . Hence α ∈ Q, α ′ = α ′ andLet K be the cubic number field Q(α) which we view as a subfield of R. Define σ : K → C to be one of the two complex embeddings, the other one being the conjugate σ . Hence α ′ = σ (α) and α ′ = σ (α). If τ is defined to be the complex conjugation, we have σ = τ • σ and σ • τ = σ .