Citation
Aminudin, Siti Syaheera and Sapar, Siti Hasana and Mohd Atan, Kamel Ariffin
(2013)
A method of estimating the padic sizes of common zeros of partial derivative polynomials associated with a complete cubic form.
In: International Conference on Mathematical Sciences and Statistics 2013 (ICMSS2013), 57 Feb. 2013, Kuala Lumpur, Malaysia. (pp. 205212).
Abstract
Let x =(x1,x2,…,xn) 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(f;q)=Σxmodqe((2if(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 derivative 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 Qp[x,y] with a complete cubic form by using Newton polyhedron technique. The polynomial is of the form f(x,y)=ax3+bx2y+cxy2+dy3+32ax2+bxy+12cy2+sx+ty+k.
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