ON THE CARDINALITY OF THE SET OF SOLUTIONS TO CONGRUENCE EQUATION ASSOCIATED WITH CUBIC FORM

Let (x) under bar =(x(1), x(2),..., x(n)) be a vector in the space Q(n) with Q field of rational numbers and q be a positive integer, f a polynomial in (x) under bar with coefficient in Q. The exponential sum associated with f is defined as S(f; q) = Sigma((x) under barq)e(2 pi if(x)/q), where the sum is taken over a complete set of residues modulo q. The value of S(f; q) depends on the estimate of cardinality vertical bar V vertical bar, the number of elements contained in the set V = {(x) under bar mod q vertical bar(f) under bar ((x) under bar) 0 mod q}, where (f) under bar ((x) under bar) is the partial derivative of (f) under bar with respect to (x) under bar. In this paper, we will discuss the cardinality of the set of solutions to congruence equation associated with a complete cubic by using Newton polyhedron technique. The polynomial is of the form f(x, y)= ax(3) + bx(2)y + cxy(2) + dy(3) + 3/2ax(2) + bxy + 1/2cy(2) + sx + ty + k.