The Convex Polytopes and Homogeneous Coordinate Rings of Bivariate Polynomials

  • Shamsatun Nahar Ahmad Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA 85000 Segamat, Johor, Malaysia
  • Nor’Aini Aris Mathematical Sciences Department, Faculty of Sciences, Universiti Teknologi Malaysia 81310 Skudai, Johor, Malaysia
  • Azlina Jumadi Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA 85000 Segamat, Johor, Malaysia

Abstract

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.

Published
2019-11-18
How to Cite
AHMAD, Shamsatun Nahar; ARIS, Nor’Aini; JUMADI, Azlina. The Convex Polytopes and Homogeneous Coordinate Rings of Bivariate Polynomials. Scientific Research Journal, [S.l.], v. 16, n. 2, p. 1-16, nov. 2019. ISSN 2289-649X. Available at: <https://myjms.mohe.gov.my/index.php/SRJ/article/view/9346>. Date accessed: 16 apr. 2024. doi: https://doi.org/10.24191/srj.v16i2.9346.