Applications of Grôbner Bases

Specialeforsvar: Mads Kofoed Madsen

Titel: Applications of Grôbner Bases

Abstract: This Thesis is a tribute to the theory of Grôbner Bases. Thus the common denominator for every theory studied throughout this paper is the Application of Grôbner Bases in problems arising in those different fields. After a succinct introduction to Grôbner Bases, the structure of Shidoku Puzzles and of the Graph Coloring Problem will first be studied, leading to Grôbner Bases being applied to solve both such puzzles and the 3-coloring problem. Afterwards, the Weak Nullstellensatz, a famous theorem of Algebraic Geometry discovered by the German, David Hilbert, is proved using Grôbner Bases, after which the theory of Automatic Theorem Proving - an approach to prove theorems of Geometry with the help of Grôbner Bases - is examined. Finally, Grôbner Bases are used to solve the Implicitization Problem, that is, finding the smallest variety containing a given parametriziation, before their Application in Invariant Theory, through showing that a given ring of invariant polynomials is isomorphic to a corresponding quotient ring, concludes this Thesis.

Vejleder: Henrik G. Holm
Censor: David Kyed, SDU