On the quartic diophantine equation f(x, y)=f(u, v)

No non-trivial solutions are known of the diophantine equation f(x, y) = f( u, v), where f(x, y) is the general quartic form given by f(x, y) = ax(4) bx(3) y + cx(2)y(2) + dxy(3) + ey(4). This paper provides a necessary and sufficient condition for the existence of non-trivial solutions of this dio phantine equation. It has also been shown that, using this condition, integ er solutions of the equation f(x, y) = f(u, v) can be obtained in specific cases. As an example, integer solutions have been obtained for the equation x(4) + x(3) y + x(2)y(2) + xy(3) + y(4) = u(4) + u(3) v + u(2)v(2) + uv(3) + v(4), which had not been solved earlier. (C) 1999 Academic Press.