Diophantine equations over global function fields I: The Thue equation

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

Diophantine Equations and Congruences over Function Fields

We generalize the methods of Pierce of counting solutions to the congruence X ≡ Y b mod D [8] and the square sieve method of counting squares in the sequence f(X) + g(Y ) [7] to the function field setting.

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.

Solving Elliptic Diophantine Equations Avoiding Thue Equations and Elliptic Logarithms

This research was supported by the Netherlands Mathematical Research Foundation SWON with nancial aid from the Netherlands Organization for Scienti c Research NWO. We determine the solutions in integers of the equation y = (x + p)(x + p) for p = 167, 223, 337, 1201. The method used was suggested to us by Yu. Bilu, and is shown to be in some cases more efficient than other general purpose method...

Thomas' Family of Thue Equations Over Imaginary Quadratic Fields