Gröbner Basis with applications

Abstract

A basis for an ideal is such that every element in the ideal can be expressed as a linear combination of the basis. With a Gröbner Basis, every polynomial can be expressed as a linear combination of the basis with a unique remainder. In recent years, there has been a growing study in such classical cases with its applications to areas outside of mathematics. We study the concept of a Gröbner Basis and analyse fundamental theorems such as Dickson’s lemma and Hilbert Basis Theorem that are necessary for the construction of the Gröbner bases and improved algorithms to produce such a basis. We provide an alternative perspective for some fundamental theorems as well as the F4 algorithm which reduces the computational complexity of Buchberger’s algorithm. Further, we explore applications of Gröbner bases to Algebraic Geometry and Commutative Algebra.