A Method of Estimating the p-adic Sizes of Common Zeros of Partial Derivative Polynomials Associated with a Seventh Degree Form

Let x = (x1, x2,...,xn) be a vector in a space Zn where Z is the ring of integers and let q be a positive integer, f a polynomial in x with coefficients in Z. The exponential sum associated with f is defined as S(f;q) = Σ exp(2πif(x)/q) where the sum is taken over a complete set of residues modulo q. The value of S(f;q) has been shown to depend on the estimate of the cardinality V , the number of elements contained in the set V={Xmodq fx≡ 0 mod q} where fx is the partial derivatives off with respect to x. To determine the cardinality of V, the information on the p-adic sizes of common zeros of the partial derivatives polynomials need to be obtained. This paper discusses a method of determining the p-adic sizes of the components of (ξ,η), a common root of partial derivatives polynomial of f(x,y) in of degree n, where n is odd based on the p-adic Newton polyhedron technique associated with the polynomial. The polynomial of degree n is of the form f(x, y) = axn + bxn-1 y + cxn-2 y2 + sx + ty + k