A new method of solving quartic and higher degree diophantine equations

Abstract: In this paper we present a new method of solving certain quartic and higher degree homogeneous polynomial diophantine equations in four variables. The method can also be extended to solve simultaneous homogeneous polynomial diophantine equations, in five or more variables, with one of the equations being of degree $\geq 4$. We show that, under certain conditions, the method yields an arbitrarily large number of integer solutions of such diophantine equations and diophantine systems, some examples being a sextic equation in four variables, a tenth degree equation in six variables, and two simultaneous equations of degrees four and six in six variables. The method of solving these homogeneous equations also simultaneously yields arbitrarily many rational solutions of certain related nonhomogeneous equations of high degree. In contrast to existing methods, we obtain the arbitrarily large number of solutions without finding a parametric solution of the equations under consideration and without relating the solutions to rational points on an elliptic curve of positive rank. It appears from the examples given in the paper that there may exist projective varieties on which there are an arbitrarily large number of integer points and on which a curve of genus 0 or 1 does not exist.