On Gröbner Bases and Their Uses in Solving System of Polynomial Equations and Graph Coloring
- 1 University of Dhaka, Bangladesh
Copyright: © 2020 Haridas Kumar Das and Nasim Reza. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
This paper is based on the analytic and computational solution procedures of Gröbner basis and its applications. We show the behavior of the ideals generated by polynomials from a polynomial ring. We also present the idea of a zero dimensional ideal and use of this ideal to solve system of polynomial equations. We then introduce an algorithmic procedure for solving a system of polynomial equations (linear and nonlinear) with a finite number of solutions extending the idea of Gröbner basis. Finally we explore the idea of Gröbner basis for coloring the vertices of a given graph. We illustrate the stated results through a number of examples. Moreover, as for auxilary and making comparison with the analytic results, we use Mathematica 9.0.1 to develop some computer algebra.
- System of Polynomial Equations
- Linear and Nonlinear
- Gröbner Basis
- Graph Coloring