Citation
Sapar, Siti Hasana and Mohd Atan, Kamel Ariffin and Mohd Said, Mohamad Rushdan
(2007)
A Method of Estimating the padic Sizes of Common Zeros of Partial Derivative Polynomials Associated with a Seventh Degree Form.
Pertanika Journal of Science & Technology, 15 (1).
pp. 6176.
ISSN 01287680
Abstract / Synopsis
Let x = (x1, x2,...,xn) be a vector in a space Zn where Z is the ring of integers and let q be a positive integer, f a polynomial in x with coefficients in Z. The exponential sum associated with f is defined as S(f;q) = Σ exp(2πif(x)/q) where the sum is taken over a complete set of residues modulo q. The value of S(f;q) has been shown to depend on the estimate of the cardinality V , the number of elements contained in the set V={Xmodq fx≡ 0 mod q} where fx is the partial derivatives off with respect to x. To determine the cardinality of V, the information on the padic sizes of common zeros of the partial derivatives polynomials need to be obtained. This paper discusses a method of determining the padic sizes of the components of (ξ,η), a common root of partial derivatives polynomial of f(x,y) in of degree n, where n is odd based on the padic Newton polyhedron technique associated with the polynomial. The polynomial of degree n is of the form f(x, y) = axn + bxn1 y + cxn2 y2 + sx + ty + k.
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