  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.
