An estimation of the p-adic sizes of common zeros of partial derivative polynomials of degree six
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 (...
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| Language: | English |
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AIP Publishing LLC
2013
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| Online Access: | http://psasir.upm.edu.my/id/eprint/57245/ http://psasir.upm.edu.my/id/eprint/57245/1/An%20estimation%20of%20the%20p-adic%20sizes%20of%20common%20zeros%20of%20partial%20derivative%20polynomials%20of%20degree%20six.pdf |
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| author | Aminudin, Siti Syaheera Sapar, Siti Hasana Mohd Atan, Kamel Ariffin |
| author_facet | Aminudin, Siti Syaheera Sapar, Siti Hasana Mohd Atan, Kamel Ariffin |
| author_sort | Aminudin, Siti Syaheera |
| building | UPM Institutional Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-15T10:51:58Z |
| format | Conference or Workshop Item |
| id | upm-57245 |
| institution | Universiti Putra Malaysia |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-15T10:51:58Z |
| publishDate | 2013 |
| publisher | AIP Publishing LLC |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | upm-572452017-09-26T03:48:38Z http://psasir.upm.edu.my/id/eprint/57245/ An estimation of the p-adic sizes of common zeros of partial derivative polynomials of degree six Aminudin, Siti Syaheera Sapar, Siti Hasana Mohd Atan, Kamel Ariffin 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. AIP Publishing LLC 2013 Conference or Workshop Item PeerReviewed application/pdf en http://psasir.upm.edu.my/id/eprint/57245/1/An%20estimation%20of%20the%20p-adic%20sizes%20of%20common%20zeros%20of%20partial%20derivative%20polynomials%20of%20degree%20six.pdf 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). 10.1063/1.4887661 |
| spellingShingle | Aminudin, Siti Syaheera Sapar, Siti Hasana Mohd Atan, Kamel Ariffin An estimation of the p-adic sizes of common zeros of partial derivative polynomials of degree six |
| title | An estimation of the p-adic sizes of common zeros of partial derivative polynomials of degree six |
| title_full | An estimation of the p-adic sizes of common zeros of partial derivative polynomials of degree six |
| title_fullStr | An estimation of the p-adic sizes of common zeros of partial derivative polynomials of degree six |
| title_full_unstemmed | An estimation of the p-adic sizes of common zeros of partial derivative polynomials of degree six |
| title_short | An estimation of the p-adic sizes of common zeros of partial derivative polynomials of degree six |
| title_sort | estimation of the p-adic sizes of common zeros of partial derivative polynomials of degree six |
| url | http://psasir.upm.edu.my/id/eprint/57245/ http://psasir.upm.edu.my/id/eprint/57245/ http://psasir.upm.edu.my/id/eprint/57245/1/An%20estimation%20of%20the%20p-adic%20sizes%20of%20common%20zeros%20of%20partial%20derivative%20polynomials%20of%20degree%20six.pdf |