A method of estimating the p-adic sizes of common zeros of partial derivative polynomials associated with a quintic form

It is known that the value of the exponential sum can be derived from the estimate of the cardinality |V|, the number of elements contained in the set where is the partial derivatives of with respect to . The cardinality of V in turn can be derived from the p-adic sizes of common zeros of the partial derivatives . 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 Zp[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) = ax5 + bx4y + cx3y2 + sx + ty + k. The estimate obtained is in terms of the p-adic sizes of the coefficients of the dominant terms in f.

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