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- W2073133030 abstract "Let K be a number field, let S be a finite set of places of K containing the archimedean places and let μ, α1, α2, α3 be non–zero elements in K. Denote by OS the ring of S–integers in K and by O × S the group of S–units. Then the set of equivalence classes (namely, up to multiplication by S–units) of the solutions (x, y, z, e1, e2, e3, e) ∈ O S × (O× S ) 4 of the diophantine equation (X − α1E1Y )(X − α2E2Y )(X − α3E3Y )Z = μE, satisfying Card{α1e1, α2e2, α3e3} = 3, is finite. With the help of this last result, we exhibit, for every integer n > 2, new families of Thue-Mahler equations of degreee n having only trivial solutions. Furthermore, we produce an effective upper bound for the number of these solutions. The proofs of this paper rest heavily on Schmidt’s subspace theorem. Resume Soit K un corps de nombres, soit S un ensemble fini de places de K contenant les places archimediennes et soient μ, α1, α2, α3 des elements non nuls de K. Notons OS l’anneau des S– entiers de K et O× S le groupe des S–unites. Alors l’ensemble des classes d’equivalence (c’est-a-dire, a multiplication pres par des S–unites) des solutions (x, y, z, e1, e2, e3, e) ∈ O S × (O× S ) 4 de l’equation diophantienne (X − α1E1Y )(X − α2E2Y )(X − α3E3Y )Z = μE, verifiant Card{α1e1, α2e2, α3e3} = 3, est fini. Grâce a ce dernier resultat, nous pouvons exhiber pour chaque entier n ≥ 3 de nouvelles familles d’equations de Thue-Mahler de degre n ne possedant que des solutions triviales. De plus, nous donnons une borne superieure explicite pour le nombre de ces solutions. Les demonstrations de cet article reposent sur le theoreme du sous–espace de Schmidt. 2010 Mathematics Subject Classification: Primary 11D59 ; Secondary 11D45 11D61 11D25 11J87" @default.
- W2073133030 created "2016-06-24" @default.
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- W2073133030 date "2012-01-01" @default.
- W2073133030 modified "2023-09-26" @default.
- W2073133030 title "Familles d'équations de Thue–Mahler n'ayant que des solutions triviales" @default.
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- W2073133030 doi "https://doi.org/10.4064/aa155-2-1" @default.
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