The Estimating of p-Adic Sizes of Common Zeros of Partial Derivative Polynomials of Degree Eight

摘要

Let x = (x_1,x_2,...,x_n) be a vector in Z~n with Z ring of integers and q be a positive integer, f a polynomial in x with coefficient in Z. The exponential sum associated with f is defined as S (f, q) = ∑_(x mod) q~e 2πif(x)/q, where the sum is taken over a complete set of residues modulo q. The value of S (f; q) depends on the estimate of 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. To determine the cardinality of V, 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 f(x, y)in Z_p [x,y] with a sixth degree form by using Newton polyhedron technique. The polynomial is of the form f(x, y) = ax~8 + bx~7y + cx~6y~2 + sx + ty + k.