Cardinality of sets associated to certain degree seven polynomials

Let f = f(x,y) be a function of two variables. Let q be an integer and let S(f;q) = ∑x mod qe 2πif(x)/q, where the sum is taken over a complete set of residue modulo q. The value of S(f;q) depends on the estimate of the cardinality |V| of the following set V={(x,y)mod q |fx, fy ≡ 0 mod q} where fx and fy are the partial derivative of f with respect to x and y. In this paper, we discuss the cardinality, |V| of the set of solutions for congruence equations of some special binary forms. Firstly we need to obtain the p-adic sizes of common zeros of the partial derivative polynomials by using Newton polyhedron technique. The polynomial that we consider is in the form of f(x, y) = ax7 + bx6y + cx5y2 + sx + ty + k.

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