On the Estimate to Solutions of Congruence Equations Associated with a Cubic Form
The set of solutions to congruence equations modulo a prime power associated with the polynomial f(x,y) = ax3 + bx2 y + cxy2 + dt + kx + my + n in Z [x,y] is examined and its cardinality estimated by employing the Newton polyhedral technique. The method involves reduction of the partial derivatives...
| Main Authors: | , |
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| Format: | Article |
| Language: | English English |
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Universiti Putra Malaysia Press
1993
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| Online Access: | http://psasir.upm.edu.my/id/eprint/3085/ http://psasir.upm.edu.my/id/eprint/3085/1/On_the_Estimate_to_Solutions_of_Congruence.pdf |
| _version_ | 1848839431257063424 |
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| author | Mohd Atan, Kamel Ariffin Abdullah, Ismail |
| author_facet | Mohd Atan, Kamel Ariffin Abdullah, Ismail |
| author_sort | Mohd Atan, Kamel Ariffin |
| building | UPM Institutional Repository |
| collection | Online Access |
| description | The set of solutions to congruence equations modulo a prime power associated with the polynomial f(x,y) = ax3 + bx2 y + cxy2 + dt + kx + my + n in Z [x,y] is examined and its cardinality estimated by employing the Newton
polyhedral technique. The method involves reduction of the partial derivatives of f to single-variable polynomials and finding 8 the determinant factor in the estimation. f x and f yare reduced to one-variable polynomials by the
employment of suitable parameters. The Newton polyhedrons associated with the polynomials so obtained are then considered. There exist common zeros
of the single-variable polynomials whose p-adic orders correspond to the intersection points in the combination of the indicator diagrams associated
with the respective Newton polyhedrons. This leads to sizes of common zeros of the partial derivatives of f. This information is then used to arrive at the above estimate. |
| first_indexed | 2025-11-15T07:11:20Z |
| format | Article |
| id | upm-3085 |
| institution | Universiti Putra Malaysia |
| institution_category | Local University |
| language | English English |
| last_indexed | 2025-11-15T07:11:20Z |
| publishDate | 1993 |
| publisher | Universiti Putra Malaysia Press |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | upm-30852013-05-27T07:05:37Z http://psasir.upm.edu.my/id/eprint/3085/ On the Estimate to Solutions of Congruence Equations Associated with a Cubic Form Mohd Atan, Kamel Ariffin Abdullah, Ismail The set of solutions to congruence equations modulo a prime power associated with the polynomial f(x,y) = ax3 + bx2 y + cxy2 + dt + kx + my + n in Z [x,y] is examined and its cardinality estimated by employing the Newton polyhedral technique. The method involves reduction of the partial derivatives of f to single-variable polynomials and finding 8 the determinant factor in the estimation. f x and f yare reduced to one-variable polynomials by the employment of suitable parameters. The Newton polyhedrons associated with the polynomials so obtained are then considered. There exist common zeros of the single-variable polynomials whose p-adic orders correspond to the intersection points in the combination of the indicator diagrams associated with the respective Newton polyhedrons. This leads to sizes of common zeros of the partial derivatives of f. This information is then used to arrive at the above estimate. Universiti Putra Malaysia Press 1993 Article PeerReviewed application/pdf en http://psasir.upm.edu.my/id/eprint/3085/1/On_the_Estimate_to_Solutions_of_Congruence.pdf Mohd Atan, Kamel Ariffin and Abdullah, Ismail (1993) On the Estimate to Solutions of Congruence Equations Associated with a Cubic Form. Pertanika Journal of Science & Technology, 1 (2). pp. 249-260. ISSN 0128-7680 English |
| spellingShingle | Mohd Atan, Kamel Ariffin Abdullah, Ismail On the Estimate to Solutions of Congruence Equations Associated with a Cubic Form |
| title | On the Estimate to Solutions of Congruence
Equations Associated with a Cubic Form |
| title_full | On the Estimate to Solutions of Congruence
Equations Associated with a Cubic Form |
| title_fullStr | On the Estimate to Solutions of Congruence
Equations Associated with a Cubic Form |
| title_full_unstemmed | On the Estimate to Solutions of Congruence
Equations Associated with a Cubic Form |
| title_short | On the Estimate to Solutions of Congruence
Equations Associated with a Cubic Form |
| title_sort | on the estimate to solutions of congruence
equations associated with a cubic form |
| url | http://psasir.upm.edu.my/id/eprint/3085/ http://psasir.upm.edu.my/id/eprint/3085/1/On_the_Estimate_to_Solutions_of_Congruence.pdf |