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# An estimation of the p-adic sizes of common zeros of partial derivative polynomials of degree six

## Citation

Aminudin, Siti Syaheera and Sapar, Siti Hasana and Mohd Atan, Kamel Ariffin (2013) An estimation of the p-adic sizes of common zeros of partial derivative polynomials of degree six. In: 21st National Symposium on Mathematical Sciences (SKSM21), 6-8 Nov. 2013, Penang, Malaysia. (pp. 622-627).

## 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={xmodq|fx≡0modq} where fx is the partial derivatives 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 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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## Additional Metadata

Item Type: Conference or Workshop Item (Paper) Faculty of ScienceInstitute for Mathematical Research https://doi.org/10.1063/1.4887661 AIP Publishing LLC Cardinality; Exponential sums; Newton polyhedron; p-adic sizes Nabilah Mustapa 26 Sep 2017 11:48 26 Sep 2017 11:48 http://www.altmetric.com/details.php?domain=psasir.upm.edu.my&doi=10.1063/1.4887661 http://psasir.upm.edu.my/id/eprint/57245 View Download Statistic

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