This is a paper in Groebner Bases and their applications. The problem sits naturally in the realm of algebraic geometry, the basics of which are introduced in this course. A Groebner Base is a certain construct derived from a similar set of equations which, because it can be realised algorithmically, enables one to do actual computations. The algorithm is implimented in computer algebra systems such as Maple. We will study the theory and applications, such as an algorithm for solving the three-colour problem in graph theory, problems in robot kinematics, integer programming, and coding theory.
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