On the cardinality of the set of solutions to congruence equation associated with cubic form

Let x = (x1, x2,..., xn) be a vector in the space ℚn with ℚ field of rational numbers and q be a positive integer, f a polynomial in x with coefficient in ℚ. The exponential sum associated with f is defined as S (f;q)=∑xmodq e 2πif(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 |V|, the number of elements contained in the set V= {x mod q |f x≡0mod q}, where fx f is the partial derivative of f with respect to x. 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)= ax3 + bx2y + cxy2 + dy3 + 3/2ax2 + bxy + 1/2cy2 + sx + ty + k.

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