A Method of Estimating the p-adic Sizes of Common Zeros of Partial Derivative Polynomials Associated with a Complete Cubic Form

Let x $=(x_{1}, x_{2}, {\ldots}, x_{n})$ be a vector in the space Q n with Q field of rational numbers and q be a positive integer, f a polynomial in x with coefficient in Q. The exponential sum associated with f is defined as $S (\textit{f}; q) = \Sigma_{x mod q}e^{((2i\textit{f}(x))/q)}$ , where the sum is taken over a complete set of residues modulo q. The value of $S (\textit{f}; q)$ depends on the estimate of cardinality $|V|$ , the number of elements contained in the set $V =\{\textit{x} mod q | \textit{f}_{\textit{x}}\equiv 0 mod q\}$ where $\textit{f}_{\textit{x}}$ is the partial derivative 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 $\textit{f}(x,y)$ in $Q_{\textit{p}}[x, y]$ with a complete cubic form by using Newton polyhedron technique. The polynomial is of the form $\textit{f}(x,y)= a\textit{x}^{3}+ b\textit{x}^{2}\textit{y} + c\textit{x}\textit{y}^{2}+d\textit{y}^{3}+ \frac{3}{2} a\textit{x}^{2}+ b\textit{x}\textit{y}+\frac{1}{2}c\textit{y}^{2}+s\textit{x}+t\textit{y}+k.$