Christian Eriksson: Gröbner Bases and Elimination in Macaulay 2

Abstract: This thesis covers fundamental concepts in algebraic geometry, polynomial elimination and Gröbner bases. It accomplishes this by introducing affine spaces and varieties to allow a better understanding of what algebraic geometry is, and then pivots to exploring Gröbner bases. This is done with the final goal of constructing an algorithm that can compute these bases deterministically. At first, monomial ideals are considered but the scope is widened to polynomial ideals. A verification of membership to Gröbner bases, the Buchberger criterion, is proven. The Buchberger algorithm, or the construction of Gröbner basis, is implemented in the computer algebra system Macaulay2 using the criterion. Univariate division, multivariate division, and solutions to systems of polynomial equations were also implemented algorithmically. The algorithms were verified programmatically with unit tests. The thesis provides an introduction to the construction and implementation of algorithms as well as a solid foundation for future expansion into algebraic geometry or construction of algorithms.