A method of estimating the p-adic sizes of common zeros of partial derivative polynomials associated with a complete cubic form
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. T...
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| Format: | Conference or Workshop Item |
| Language: | English |
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Springer Singapore
2013
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| Online Access: | http://psasir.upm.edu.my/id/eprint/41157/ http://psasir.upm.edu.my/id/eprint/41157/1/A%20method%20of%20estimating%20the%20p-adic%20sizes%20of%20common%20zeros%20of%20partial%20derivative%20polynomials%20associated%20with%20a%20complete%20cubic%20form.pdf |
| _version_ | 1848849620069777408 |
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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 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={xmodq|fx≡0modq} where fx is the partial derivative 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 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. |
| first_indexed | 2025-11-15T09:53:17Z |
| format | Conference or Workshop Item |
| id | upm-41157 |
| institution | Universiti Putra Malaysia |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-15T09:53:17Z |
| publishDate | 2013 |
| publisher | Springer Singapore |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | upm-411572017-08-24T08:37:25Z http://psasir.upm.edu.my/id/eprint/41157/ A method of estimating the p-adic sizes of common zeros of partial derivative polynomials associated with a complete cubic form Aminudin, Siti Syaheera Sapar, Siti Hasana Mohd Atan, Kamel Ariffin 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={xmodq|fx≡0modq} where fx is the partial derivative 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 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. Springer Singapore 2013 Conference or Workshop Item PeerReviewed application/pdf en http://psasir.upm.edu.my/id/eprint/41157/1/A%20method%20of%20estimating%20the%20p-adic%20sizes%20of%20common%20zeros%20of%20partial%20derivative%20polynomials%20associated%20with%20a%20complete%20cubic%20form.pdf Aminudin, Siti Syaheera and Sapar, Siti Hasana and Mohd Atan, Kamel Ariffin (2013) A method of estimating the p-adic sizes of common zeros of partial derivative polynomials associated with a complete cubic form. In: International Conference on Mathematical Sciences and Statistics 2013 (ICMSS2013), 5-7 Feb. 2013, Kuala Lumpur, Malaysia. (pp. 205-212). 10.1007/978-981-4585-33-0_21 |
| spellingShingle | Aminudin, Siti Syaheera Sapar, Siti Hasana Mohd Atan, Kamel Ariffin A method of estimating the p-adic sizes of common zeros of partial derivative polynomials associated with a complete cubic form |
| title | A method of estimating the p-adic sizes of common zeros of partial derivative polynomials associated with a complete cubic form |
| title_full | A method of estimating the p-adic sizes of common zeros of partial derivative polynomials associated with a complete cubic form |
| title_fullStr | A method of estimating the p-adic sizes of common zeros of partial derivative polynomials associated with a complete cubic form |
| title_full_unstemmed | A method of estimating the p-adic sizes of common zeros of partial derivative polynomials associated with a complete cubic form |
| title_short | A method of estimating the p-adic sizes of common zeros of partial derivative polynomials associated with a complete cubic form |
| title_sort | method of estimating the p-adic sizes of common zeros of partial derivative polynomials associated with a complete cubic form |
| url | http://psasir.upm.edu.my/id/eprint/41157/ http://psasir.upm.edu.my/id/eprint/41157/ http://psasir.upm.edu.my/id/eprint/41157/1/A%20method%20of%20estimating%20the%20p-adic%20sizes%20of%20common%20zeros%20of%20partial%20derivative%20polynomials%20associated%20with%20a%20complete%20cubic%20form.pdf |