An Estimating the p-adic Sizes of Common Zeros of Partial Derivative Polynomials

Let v be a vector in the space with ring of integers and be a positive integer, a polynomial in with coefficient in . The exponential sum associated with is defined as , where the sum is taken over a complete set of residues modulo . The value of depend on the estimate of cardinality , the number of elements contained in the set where is the partial derivatives of with respect to . To determine the cardinality of , the p-adic sizes of common zeros of the partial derivative polynomials need to be obtained. In this paper we estimate the p-adic sizes of common zeros of partial derivative polynomials of in of degree nine by using Newton polyhedron technique. The degree nine polynomial is of the form .