Estimation of p-Adic Sizes of Common Zeroes of Partial Derivative Polynomials Associated with a Quintic Form

Let x ={xi,x2,...,xn}be a vector in a space Zn with Z ring of integers and let q bea positive integer, f a polynomial in x with coefficients in Z. The exponential sum associated with f is defined as S(f;q) = Σexp(2(pi)if (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 ={x mod q | fx =0 mod q} where fx is thepartial derivatives of / with respect to x . To determine the cardinality of V, the information on the jb-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(xi,eta) a common root of partial derivative polynomials of f[x, y) in Zp[x, y] of degree five based on the jfr-adic Newton polyhedron technique associated with the polynomial. The quintic polynomial is of the form f(x,y) = ax5 + bx4y + cx3y + dx2 y3 + exy4 + my5 + nx + ty + k.