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Buchbergers algorithm for computing Groebner Basis

Polynomial bases are a generalization of systems of linear equations. If we have a system of linear equations, say -

In Gaussian elimination, we take a system of linear equations and convert them to a basis where the solution becomes obvious.

A framework is provided for defining polynomial ideals. The various objects that are a part of this solution form a hierarchical structure:

PolynomialBasis -> Polynomial -> Monomial

The code starts with the building blocks like LCM, polynomial division, S-polynomials and builds up to Groebner basis. Some applications of Groebner basis are also provided.

You can execute the code via Program.cs.

All algorithms are based on the book - Ideals, Varieties and Algorithms by Cox, Little and O'Shea

In comments and documentation, "CLO" referes to this book (Cox, Little and O'Shea). I wrote a blog on this topic providing some background.