Citation
Aminudin, Siti Syaheera and Sapar, Siti Hasana and Mohd Atan, Kamel Ariffin
(2013)
An estimation of the padic sizes of common zeros of partial derivative polynomials of degree six.
In: 21st National Symposium on Mathematical Sciences (SKSM21), 68 Nov. 2013, Penang, Malaysia. (pp. 622627).
Abstract / Synopsis
Let x = (x1,x2,...,xn) be a vector in Zn 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) = ∑xmodqe((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={xmodqfx≡0modq} where fx is the partial derivatives of f with respect to x. To determine the cardinality of V, the padic sizes of common zeros of the partial derivative polynomials need to be obtained. In this paper, we estimate the padic sizes of common zeros of partial derivative polynomials of f(x,y) in Zp[x,y] with a sixth degree form by using Newton polyhedron technique. The polynomial is of the form f(x,y) = ax6+bx5y+cx4y2+sx+ty+k.
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