Thomas’ conjecture over function fields

Ziegler, Volker

Journal de théorie des nombres de Bordeaux, Tome 19 (2007) no. 1, pp. 289-309.

La conjecture de Thomas affirme que, pour des polynômes unitaires $p_1,...,p_d\in \mathbb{Z}\left[a\right]$ tels que $0<deg{p}_1<\cdots <deg{p}_d$, l’équation de Thue

$$(X-p_1\left(a\right)Y)\cdots (X-p_d\left(a\right)Y)+Y^d=1$$

n’admet pas de solution non triviale (dans les entiers relatifs) pourvu que $a\ge a_0$, avec une borne effective $a_0$. Nous nous intéressons à un analogue de la conjecture de Thomas sur les corps de fonctions pour le degré $d=3$ et en donnons un contrexemple.

Thomas’ conjecture is, given monic polynomials $p_1,$ $...,p_d\in \mathbb{Z}\left[a\right]$ with $0<deg{p}_1<\cdots <deg{p}_d$, then the Thue equation (over the rational integers)

$$(X-p_1\left(a\right)Y)\cdots (X-p_d\left(a\right)Y)+Y^d=1$$

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 $d=3$.

[1] A. Baker, Linear forms in the logarithms of algebraic numbers. I, II, III. Mathematika 13 (1966), 204–216; ibid. 14 (1967), 102–107; ibid. 14 (1967), 220–228. | MR | Zbl

[2] A. Baker, Contributions to the theory of Diophantine equations. I. On the representation of integers by binary forms. Philos. Trans. Roy. Soc. London Ser. A 263 (1967/1968), 173–191, | MR | Zbl

[3] A. Baker, Linear forms in the logarithms of algebraic numbers. IV. Mathematika 15 (1968), 204–216. | MR | Zbl

[4] P. M. Cohn, Algebraic numbers and algebraic functions. Chapman and Hall Mathematics Series. Chapman & Hall, London, 1991. | MR | Zbl

[5] C. Fuchs, V. Ziegler, On a family of Thue equations over function fields. Monatsh. Math. 147 (2006), 11–23. | MR | Zbl

[6] C. Fuchs, V. Ziegler, Thomas’ family of Thue equations over function fields. Quart. J. Math. 57 (2006), 81–91. | Zbl

[7] B. P. Gill, An analogue for algebraic functions of the Thue-Siegel theorem. Ann. of Math. (2) 31(2) (1930), 207–218. | MR

[8] C. Heuberger, On a conjecture of E. Thomas concerning parametrized Thue equations. Acta. Arith. 98 (2001), 375–394. | MR | Zbl

[9] S. Lang, Elliptic curves: Diophantine analysis. Volume 231 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1978. | MR | Zbl

[10] E. Lee, Studies on Diophantine Equations. PhD thesis, Cambridge University, 1992.

[11] G. Lettl, Thue equations over algebraic function fields. Acta Arith. 117(2) (2005), 107–123. | MR | Zbl

[12] R. C. Mason, On Thue’s equation over function fields. J. London Math. Soc. (2) 24(3) (1981), 414–426. | Zbl

[13] R. C. Mason, The hyperelliptic equation over function fields. Math. Proc. Camb. Philos. Soc. 93 (1983), 219–230. | MR | Zbl

[14] R. C. Mason, Diophantine equations over function fields. Volume 96 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1984. | MR | Zbl

[15] M. Mignotte, N. Tzanakis, On a family of cubics. J. Number Theory 39(1) (1991), 41–49. | MR | Zbl

[16] P. Ribenboim, Remarks on existentially closed fields and diophantine equations. Rend. Sem. Mat. Univ. Padova 71 (1984), 229–237. | Numdam | MR | Zbl

[17] M. Rosen, Number theory in function fields. Volume 210 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2002. | MR | Zbl

[18] W. M. Schmidt, Thue’s equation over function fields. J. Austral. Math. Soc. Ser. A 25(4) (1978), 385–422. | Zbl

[19] E. Thomas, Solutions to certain families of Thue equations. J. Number Theory 43(3) (1993), 319–369. | MR | Zbl

[20] A. Thue, Über Annäherungswerte algebraischer Zahlen. J. Reine und Angew. Math. 135 (1909), 284–305.

[21] E. Wirsing, Approximation mit algebraischen Zahlen beschränkten Grades. J. Reine Angew. Math. 206 (1960), 67–77. | MR | Zbl

[22] V. Ziegler, On a family of Cubics over Imaginary Quadratic Fields. Period. Math. Hungar. 51(2) (2005), 109–130. | MR | Zbl