A METHOD OF ESTIMATING THE p-ADIC SIZES OF COMMON ZEROS OF PARTIAL DERIVATIVE POLYNOMIALS ASSOCIATED WITH A QUINTIC FORM

摘要

It is known that the value of the exponential sum S(f;q) = ∑_(x mod q) exp(2πif/q) can be derived from the estimate of the cardinality |V|, the number of elements contained in the set V = {x mod q|f_x ≡ 0 mod q} where f_x is the partial derivatives of f with respect to x. The cardinality of V in turn can be derived from the p-adic sizes of common zeros of the partial derivatives f_x . This paper presents a method of determining the p-adic sizes of the components of (ξ, η) a common root of partial derivative polynomials of f(x,y) in Z_p[x,y] of degree five based on the p-adic Newton polyhedron technique associated with the polynomial. The degree five polynomial is of the form f(x,y) = ax~5 + bx~4y + cx~3y~2 + sx + ty + k. The estimate obtained is in terms of the p-adic sizes of the coefficients of the dominant terms in f.