Let Q(1), Q(2) is an element of Z [X, Y, Z] be two ternary quadratic forms and u(1), u(2) is an element of Z. In this paper we consider the problem of solving the system of equations Q(1)(x, y, z)=u(1), (1) Q(2)(x, y, z)=u(2) in x, y, z is an element of Z with gcd(x, y, z,)=1. According to Mordell [12] the coprime solutions of Q(o)(x, y, z)=u(1)Q(2)(x, y, z)-u(2)Q(1)(x, y, z)=0 can be presented by finitely many expressions of the form x=f(x)(p, q), y=f(y)(p, q), z=f(z)(p, q), where f(x),f(y),f(z) is an element of Z [P, Q] are binary quadratic forms and p, q are coprime integers. Substituting these expressions into one of the equations of (1), we obtain a quartic homogeneous equation in two variables. If it is irreducible it is a quartic Thue equation, otherwise it can be solved even easier. The finitely many solutions p, q of that equation then yield all solutions x, y, z of (1). We also discuss two applications. In [8] we showed that the problem of solving index form equations in quartic numbers fields K can be reduced to the resolution of a cubic equation F(u, v)=i and a corresponding system of quadratic equations Q(1)(x, y, z)=u, Q(2)(x, y, z)=v, where F is a binary cubic form and Q(1), Q(2) are ternary quadratic forms. In this case the application of the new ideas for the resolution of (1) leads to quartic Thue equations which split over the same quartic field K. The second application is to the calculation of all integral points of an elliptic curve. (C) 1996 Academic Press, Inc.
