Thue equations and CM-fields

Thue equations and CM-fields

We obtain a polynomial type upper bound for the size of the integral solutions of Thue equations F (X,Y ) = b defined over a totally real number field K, assuming that F (X, 1) has a root α such that K(α) is a CM-field. Furthermore, we give an algorithm for the computation of the integral solutions of such an equation.

On Thue equations of splitting type over function fields

In this paper we consider Thue equations of splitting type over the ring k[T ], i.e. they have the form X(X − p1Y ) · · · (X − pd−1Y )− Y d = ξ, with p1, . . . , pd−1 ∈ k[T ] and ξ ∈ k. In particular we show that such Thue equations have only trivial solutions provided the degree of pd−1 is large, with respect to the degree of the other parameters p1, . . . , pd−2.

Thomas' Family of Thue Equations Over Imaginary Quadratic Fields

Thue Equations and Elliptic Curves

We discuss estimates for the number of solutions of Thue equations and for the number of twists of elliptic curves over the rationals with rank at least 2. We indicate some of the connections between these problems. 1. THUE EQUATIONS Let F be a binary form with rational integer coefficients and with r ≥ 3. Let h be a non-zero integer. In 1909, Thue [43] proved that if F is irreducible then the ...

Diophantine Equations over Global Function Fields II: R-Integral Solutions of Thue Equations