Quartic Thue equations (Q1038480)

The two main results of this paper are the following: Theorem 1. Let \(F(x,y)\) be an irreducible quartic binary form with integral coefficients. The Diophantine equation \(|F(x,y)|=1\) has at most 61 integral solutions in \(x\) and \(y\) (with \((x,y)\) and \((-x,-y)\) regarded as the same) provided that the discriminant of \(F\) is greater than \(D_0\), where \(D_0\) is an effectively absolute constant. Theorem 2. Let \(F(x,y)\) be as above and assume that it splits over the real numbers. Then the Diophantine equation \(|F(x,y)|=1\) has at most 37 integral solutions in \(x\) and \(y\) (with \((x,y)\) and \((-x,-y)\) regarded as the same) provided that the discriminant of \(F\) is greater than \(D_1\), where \(D_1\) is an effectively absolute constant. The proof combines a detailed study of ``large'' and ``small'' solutions.