The convex polytopes and homogeneous coordinate rings of bivariate polynomials / Shamsatun Nahar Ahmad, Nor’Aini Aris and Azlina Jumadi
Concepts from algebraic geometry such as cones and fans are related to toric varieties and can be applied to determine the convex polytopes and homogeneous coordinate rings of multivariate polynomial systems. The homogeneous coordinates of a system in its projective vector space can be associated wi...
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| Format: | Article |
| Language: | English |
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Penerbit UiTM (UiTM Press)
2019
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| Subjects: | |
| Online Access: | https://ir.uitm.edu.my/id/eprint/28260/1/28260.pdf https://doi.org/10.24191/srj.v16i2.5507 |
| _version_ | 1796902665731440640 |
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| author | Ahmad, Shamsatun Nahar Aris, Nor’Aini Jumadi, Azlina |
| author_facet | Ahmad, Shamsatun Nahar Aris, Nor’Aini Jumadi, Azlina |
| author_sort | Ahmad, Shamsatun Nahar |
| collection | UITM |
| description | Concepts from algebraic geometry such as cones and fans are related to toric varieties and can be applied to determine the convex polytopes and homogeneous coordinate rings of multivariate polynomial systems. The homogeneous coordinates of a system in its projective vector space can be associated with the entries of the resultant matrix of the system under consideration. This paper presents some conditions for the homogeneous coordinates of a certain system of bivariate polynomials through the construction and implementation of the Sylvester-Bèzout hybrid resultant matrix formulation. This basis of the implementation of the Bèzout block applies a combinatorial approach on a set of linear inequalities, named 5-rule. The inequalities involved the set of exponent vectors of the monomials of the system and the entries of the matrix are determined from the coefficients of facets variable known as brackets. The approach can determine the homogeneous coordinates of the given system and the entries of the Bèzout block. Conditions for determining the homogeneous coordinates are also given and proven. |
| first_indexed | 2024-03-06T02:08:21Z |
| format | Article |
| id | uitm.eprints-8260 |
| institution | Universiti Teknologi MARA |
| language | English |
| last_indexed | 2024-03-06T02:08:21Z |
| publishDate | 2019 |
| publisher | Penerbit UiTM (UiTM Press) |
| record_format | dspace |
| spelling | uitm.eprints-82602020-02-11T09:59:48Z https://ir.uitm.edu.my/id/eprint/28260/ The convex polytopes and homogeneous coordinate rings of bivariate polynomials / Shamsatun Nahar Ahmad, Nor’Aini Aris and Azlina Jumadi srj Ahmad, Shamsatun Nahar Aris, Nor’Aini Jumadi, Azlina Algebra Concepts from algebraic geometry such as cones and fans are related to toric varieties and can be applied to determine the convex polytopes and homogeneous coordinate rings of multivariate polynomial systems. The homogeneous coordinates of a system in its projective vector space can be associated with the entries of the resultant matrix of the system under consideration. This paper presents some conditions for the homogeneous coordinates of a certain system of bivariate polynomials through the construction and implementation of the Sylvester-Bèzout hybrid resultant matrix formulation. This basis of the implementation of the Bèzout block applies a combinatorial approach on a set of linear inequalities, named 5-rule. The inequalities involved the set of exponent vectors of the monomials of the system and the entries of the matrix are determined from the coefficients of facets variable known as brackets. The approach can determine the homogeneous coordinates of the given system and the entries of the Bèzout block. Conditions for determining the homogeneous coordinates are also given and proven. Penerbit UiTM (UiTM Press) 2019 Article PeerReviewed text en https://ir.uitm.edu.my/id/eprint/28260/1/28260.pdf The convex polytopes and homogeneous coordinate rings of bivariate polynomials / Shamsatun Nahar Ahmad, Nor’Aini Aris and Azlina Jumadi. (2019) Scientific Research Journal <https://ir.uitm.edu.my/view/publication/Scientific_Research_Journal/>, 16 (2). pp. 1-16. ISSN 1675-7009 https://srj.uitm.edu.my/ https://doi.org/10.24191/srj.v16i2.5507 https://doi.org/10.24191/srj.v16i2.5507 |
| spellingShingle | Algebra Ahmad, Shamsatun Nahar Aris, Nor’Aini Jumadi, Azlina The convex polytopes and homogeneous coordinate rings of bivariate polynomials / Shamsatun Nahar Ahmad, Nor’Aini Aris and Azlina Jumadi |
| title | The convex polytopes and homogeneous coordinate rings of bivariate polynomials / Shamsatun Nahar Ahmad, Nor’Aini Aris and Azlina Jumadi |
| title_full | The convex polytopes and homogeneous coordinate rings of bivariate polynomials / Shamsatun Nahar Ahmad, Nor’Aini Aris and Azlina Jumadi |
| title_fullStr | The convex polytopes and homogeneous coordinate rings of bivariate polynomials / Shamsatun Nahar Ahmad, Nor’Aini Aris and Azlina Jumadi |
| title_full_unstemmed | The convex polytopes and homogeneous coordinate rings of bivariate polynomials / Shamsatun Nahar Ahmad, Nor’Aini Aris and Azlina Jumadi |
| title_short | The convex polytopes and homogeneous coordinate rings of bivariate polynomials / Shamsatun Nahar Ahmad, Nor’Aini Aris and Azlina Jumadi |
| title_sort | convex polytopes and homogeneous coordinate rings of bivariate polynomials shamsatun nahar ahmad nor aini aris and azlina jumadi |
| topic | Algebra |
| url | https://ir.uitm.edu.my/id/eprint/28260/1/28260.pdf https://doi.org/10.24191/srj.v16i2.5507 |
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