A Method of Estimating the p-adic Sizes of Common Zeros of Partial Derivative Polynomials Associated with a Seventh Degree Form
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...
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| Format: | Article |
| Language: | English |
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Universiti Putra Malaysia Press
2007
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| Online Access: | http://psasir.upm.edu.my/id/eprint/11966/ |
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| author | Sapar, Siti Hasana Mohd Atan, Kamel Ariffin Mohd Said, Mohamad Rushdan |
| author_facet | Sapar, Siti Hasana Mohd Atan, Kamel Ariffin Mohd Said, Mohamad Rushdan |
| author_sort | Sapar, Siti Hasana |
| building | UPM Institutional Repository |
| collection | Online Access |
| description | 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 p-adic sizes of common zeros of the partial derivatives polynomials need to be obtained. This paper discusses a method of determining the p-adic 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 p-adic Newton polyhedron technique associated with the polynomial. The polynomial of degree n is of the form f(x, y) = axn + bxn-1 y + cxn-2 y2 + sx + ty + k. |
| first_indexed | 2025-11-15T07:47:41Z |
| format | Article |
| id | upm-11966 |
| institution | Universiti Putra Malaysia |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-15T07:47:41Z |
| publishDate | 2007 |
| publisher | Universiti Putra Malaysia Press |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | upm-119662011-04-21T08:32:19Z http://psasir.upm.edu.my/id/eprint/11966/ A Method of Estimating the p-adic Sizes of Common Zeros of Partial Derivative Polynomials Associated with a Seventh Degree Form Sapar, Siti Hasana Mohd Atan, Kamel Ariffin Mohd Said, Mohamad Rushdan 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 p-adic sizes of common zeros of the partial derivatives polynomials need to be obtained. This paper discusses a method of determining the p-adic 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 p-adic Newton polyhedron technique associated with the polynomial. The polynomial of degree n is of the form f(x, y) = axn + bxn-1 y + cxn-2 y2 + sx + ty + k. Universiti Putra Malaysia Press 2007 Article PeerReviewed Sapar, Siti Hasana and Mohd Atan, Kamel Ariffin and Mohd Said, Mohamad Rushdan (2007) A Method of Estimating the p-adic Sizes of Common Zeros of Partial Derivative Polynomials Associated with a Seventh Degree Form. Pertanika Journal of Science & Technology, 15 (1). pp. 61-76. ISSN 0128-7680 English |
| spellingShingle | Sapar, Siti Hasana Mohd Atan, Kamel Ariffin Mohd Said, Mohamad Rushdan A Method of Estimating the p-adic Sizes of Common Zeros of Partial Derivative Polynomials Associated with a Seventh Degree Form |
| title | A Method of Estimating the p-adic Sizes of Common Zeros of Partial Derivative Polynomials Associated with a Seventh Degree Form |
| title_full | A Method of Estimating the p-adic Sizes of Common Zeros of Partial Derivative Polynomials Associated with a Seventh Degree Form |
| title_fullStr | A Method of Estimating the p-adic Sizes of Common Zeros of Partial Derivative Polynomials Associated with a Seventh Degree Form |
| title_full_unstemmed | A Method of Estimating the p-adic Sizes of Common Zeros of Partial Derivative Polynomials Associated with a Seventh Degree Form |
| title_short | A Method of Estimating the p-adic Sizes of Common Zeros of Partial Derivative Polynomials Associated with a Seventh Degree Form |
| title_sort | method of estimating the p-adic sizes of common zeros of partial derivative polynomials associated with a seventh degree form |
| url | http://psasir.upm.edu.my/id/eprint/11966/ |