Abstract :
No non-trivial solutions are known of the diophantine equationf(x, y)=f(u, v), wheref(x, y) is the general quartic form given byf(x, y)=ax4+bx3y+cx2y2+dxy3+ey4. This paper provides a necessary and sufficient condition for the existence of non-trivial solutions of this diophantine equation. It has also been shown that, using this condition, integer solutions of the equationf(x, y)=f(u, v) can be obtained in specific cases. As an example, integer solutions have been obtained for the equationx4+x3y+x2y2+xy3+y4=u4+u3v+u2v2+uv3+v4,which had not been solved earlier.
