A Systematic Study of Groebner Basis Methods

TL;DR

This comprehensive study explores the theory, generalizations, and applications of Groebner Bases across various algebraic structures, emphasizing their utility in solving polynomial and function ring problems.

Contribution

It provides a systematic overview of Groebner Bases, including new generalizations and applications in function rings and modules over different rings.

Findings

Extended the theory of Groebner Bases to function rings.

Demonstrated applications in elimination theory and polynomial mappings.

Unified various algebraic structures under Groebner Bases framework.

Abstract

This habilitation (German variant of a PhD on top of a PhD) thesis presents the quintessence of the ideas and experiences with Groebner Bases of Birgit Reinert. She died unexpectedly without providing an abstract. As arXiv requires an abstract, let us quote from the table of contents: History of Groebner Bases. The two definitions of Groebner Bases. Generalizations. Groebner Bases in Function Rings -- A Guide for Introducing Relations to Algebraic Structures. Applications of Groebner Bases to Function Rings. Groebner Bases in Polynomial Rings. Reduction Rings. Quotients of Reduction Rings. Sums of Reduction Rings. Modules over Reduction Rings. Polynomial Rings over Reduction Rings. Function Rings: Right Ideals and Right Standard Representations. Function Rings over Fields. Function Rings over Reduction Rings. Function Rings over the Integers. Right F-Modules. Ideals and Standard Representations. Two-sided Modules. Applications of Groebner Bases: Natural Applications. Quotient Rings. Elimination Theory. Polynomial Mappings. Systems of One-sided Equations in Function Rings over the Integers.