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

Let ( , ,..., ) 1 2 n x = x x x 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) = Σexp(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 = 0modq} where fx is the partial derivatives of f 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−1y + cxn−2 y2 + sx + ty + k