Computational Commutative Algebra and Geometry
Introduction
Gröbner basis is a specific generating set of an algebraic structure called as ideal in multivariable polynomials. It has interesting properties connecting computational algebraic geometry and computational commutative algebra, and also having applications in computer algebra. The most important application being “solving any system of polynomial equations”. Gröbner basis is a generalization of the idea of greatest common divisors of multivariable polynomials, and the row-reduced echelon form of a system of linear equations. This project explores the theory relating commutative algebra and algebraic geometry as well as the applications of Gröbner basis.
The report is divided into parts: An Introduction to Algebra and Geometry followed by An Introduction to Gröbner basis then Applications of Gröbner basis. We start with an introduction to algebraic concepts, focussing over polynomials, ideals and their algorithms. Then we study geometric concepts such as varieties and their relationship with ideals which is probed with interesting problem. One such problem of “Ideal Membership” is broken down into simpler cases and the development of Gröbner basis Theory and Computation is motivated using it. Then, we focus on select applications of Gröbner basis from countless many available in literature and also discuss SageMath implementations for some of them. I have tried to make this report interesting and also covered fundamentals. Still, this is just a glimpse of an extensive topic like Gröbner basis.
The report can be viewed here.
The slides can be viewed here.
Applications
- System of Linear Equations Symbolic Solver
- System of Polynomial Equations Solver
- Lagrange Multipliers One Constraint and Generalized.
- Sudoku Solver
TeX-nical Details
This is the primary TeX file which is to be compiled to get the report.
This is the primary TeX file which is to be compiled to get the slides.
This is the BibTeX file containing the references used by me.