Effective results for unit points on curves over finitely generated domains

Abstract

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.

Effective results for division points on curves in \mathbb{G}_m^2

“…Our result is not only effective, but also quantitative in the sense that an upper bound for the sizes of the solutions x, y ∈ Γ is provided. The presented result is a common generalization of the results of Bombieri and Gubler [7, p. 147, Theorem 5.4.5], Bérczes, Evertse, Győry and Pontreau [5] and that of Bérczes [2]. Further, our result is also a generalization of the result of Bérczes, Evertse and Győry [3] and of Evertse and Győry [12] on unit equations, since taking F (X, Y ) = aX + bY − 1 in the main result of the present paper we just get an effective finiteness result for generalized unit equations in two unknowns over the division group of an arbitrary finitely generated group.…”

“…This is Proposition 6.1 of [2], however, with a slightly different bound then it was originally obtained in [5]. …”

“…We mention that this condition is effectively decidable, as is explained in Section 3.1 of [2]. Let N := deg F denote the total degree of F , and assume that F is given by specifying a representativeã ij of its coefficient a ij for each (i, j) ∈ I.…”

“…Later Bérczes, Evertse and Győry [4] obtained effective results for Thue equations, hyper-and superelliptic equations and for the Schinzel-Tijdeman equation over arbitrary finitely generated domains. Lastly Bérczes in [2] proved an effective result for equations F (x, y) = 0 in x, y ∈ A * for arbitrary finitely generated domains A, thus giving an effective version of the below result of Lang [15].…”

“…The first effective versions of this result of Lang have been proved by Bombieri and Gubler [7, p. 147, Theorem 5.4.5] (see Bérczes, Evertse Győry and Pontreau [5] for an explicit version) for the number field case and by Bérczes [2] in its full generality, over finitely generated domains. We mention that the effective results are proved under a slightly stronger condition than (1.2), namely in (1.2) α ∈ K is assumed instead of α ∈ Γ. Denote by Γ the division group of Γ, i.e.…”