A Method of Estimating the (rho)-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 with Z ring of integers and let qbe a positive integer, fa polynomial in x with coefficients in Z. The exponential sum associated with f is defined as S(j;q) = ∑exp(2пif(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 IVl, the number of elements contained in the set V = {xmodq I fx (equiv) 0modq} where f x the partial derivative off f with respect to x. To determine the cardinality of v: the information on the (rho)-adic sizes of common zeros of the partial derivatives polynomials need to be obtained. This paper discusses a method of determining the (rho)-adic sizes of the components of (xi,eta) a common root of partial derivatives polynomial of f(x,y) in Z(rho)[x,y] of degree seven based on the (rho)-adic Newton polyhedron technique associated with the polynomial. The seventh degree form considered is of the type f(x,y) = ax7 + bx^6y + cx^5y^3 + sx + ty + k.