Twisted Thue equations with multiple exponents in fixed number fields

Hilgart, Tobias; Ziegler, Volker

Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 2, pp. 621-635

Let $K$ be a number field of degree $d\ge 3$ and fix $s\le d-2$ multiplicatively independent ${\gamma }_1,\cdots ,{\gamma }_s\in K^{*}$ that fulfil some technical requirements, which can be vastly simplified to $\mathbb{Q}$-linearly independence, given Schanuel’s conjecture. We then consider the twisted Thue equation

$$\left|N_{K/\mathbb{Q}}(X-{\gamma }_1^{t_1}\cdots {\gamma }_s^{t_s}Y)\right|=1,$$

and prove that it has only finitely many solutions $(x,y;t_1,\cdots ,t_s)$ in ${\mathbb{Z}}^2\times {\mathbb{N}}^s$ with $xy\ne 0$ and $\mathbb{Q}({\gamma }_1^{t_1}\cdots {\gamma }_s^{t_s})=K$, all of which are effectively computable.

Soit $K$ un corps de nombres de degré $d\ge 3.$ On fixe $s\le d-2$ éléments multiplicativement indépendants et remplissant certaines conditions techniques, qui se réduisent à une condition d’indépendance $\mathbb{Q}$-linéaire si on admet la conjecture de Schanuel. Nous considérons l’équation de Thue tordue

$$\left|N_{K/\mathbb{Q}}(X-{\gamma }_1^{t_1}\cdots {\gamma }_s^{t_s}Y)\right|=1,$$

et prouvons qu’il n’existe qu’un nombre fini de solutions $(x,y;t_1,\cdots ,t_s)$ dans ${\mathbb{Z}}^2\times {\mathbb{N}}^s$ avec $xy\ne 0$ et $\mathbb{Q}({\gamma }_1^{t_1}\cdots {\gamma }_s^{t_s})=K$. Ces solutions sont effectivement calculables.

Classification : 11N56, 14G42
Keywords: Multiplicative and norm form equations, Exponential Diophantine equations

[1] Baker, Alan; Wüstholz, Gisbert Logarithmic forms and group varieties, J. Reine Angew. Math., Volume 442 (1993), pp. 19-62 | DOI | Zbl | MR

[2] Bombieri, Enrico On the Thue–Siegel–Dyson theorem, Acta Math., Volume 148 (1982), pp. 255-296 | Zbl | DOI

[3] Bugeaud, Yann; Győry, Kálmán Bounds for the solutions of Thue–Mahler equations and norm form equations, Acta Arith., Volume 74 (1996) no. 3, pp. 273-292 | Zbl | DOI | MR

[4] Lang, Serge Introduction to Transcendental Numbers, Addison-Wesley Publishing Group, 1966 | MR

[5] Levesque, Claude; Waldschmidt, Michel Familles d’équations de Thue-Mahler n’ayant que des solutions triviales, Acta Arith., Volume 155 (2012) no. 2, pp. 117-138 | DOI | Zbl | MR

[6] Levesque, Claude; Waldschmidt, Michel Solving effectively some families of Thue Diophantine equations, Mosc. J. Comb. Number Theory, Volume 3 (2013) no. 3-4, pp. 118-144 | Zbl | MR

[7] Levesque, Claude; Waldschmidt, Michel A family of Thue equations involving powers of units of the simplest cubic fields, J. Théor. Nombres Bordeaux, Volume 27 (2015) no. 2, pp. 537-563 | Numdam | Zbl | DOI | MR

[8] Smyth, Chris The Mahler measure of algebraic numbers: a survey, Number Theory and Polynomials (McKee, James; Smyth, Chris, eds.) (London Mathematical Society Lecture Note Series), Cambridge University Press, 2008, pp. 322-349 | Zbl | DOI

[9] Thomas, Emery Complete solutions to a family of cubic Diophantine equations, J. Number Theory, Volume 34 (1990) no. 2, pp. 235-250 | Zbl | DOI | MR

[10] Tijdeman, Robert On integers with many small prime factors, Compos. Math., Volume 26 (1973) no. 3, pp. 319-330 | Numdam | Zbl | MR