Year

2024

Season

Summer

Paper Type

Master's Thesis

College

College of Arts and Sciences

Degree Name

Master of Science in Mathematical Sciences (MS)

Department

Mathematics & Statistics

NACO controlled Corporate Body

University of North Florida. Department of Mathematics and Statistics

First Advisor

Dr. Daniel Dreibelbis

Second Advisor

Dr. Scott Hockwald

Third Advisor

Dr. Jae-Ho Lee

Department Chair

Dr. Richard Patterson

College Dean

Dr. Kaveri Subrahmanyam

Abstract

Grobner bases are essential tools in algebraic geometry, used to simplify and solve systems of polynomial equations. These bases revolutionized computational methods in various branches of mathematics after being introduced in 1965 by Bruno Buchberger. This thesis explores the foundational concepts of Grobner bases, including their formation and properties. It also demonstrates their use in solving mathematical problems in algebraic geometry, including the ideal membership problem. As an application, we show how Grobner bases can be used to determine whether a polynomial mapping is tame. This concept is crucial for analyzing the topology near singular points and establishing whether a real polynomial mapping has a Milnor fibration. The insights gained from this study highlight the significant theoretical and practical contributions of Grobner bases to the field of algebraic geometry.

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