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The purpose of this book is twofold: to present some basic ideas in
commutative algebra and algebraic geometry and to introduce topics
of current research, centered around the themes of Groebner bases,
resultants and syzygies. The presentation of the material combines
definitions and proofs with an emphasis on concrete examples. The
authors illustrate the use of software such as Mathematica and
Singular. The design of the text in each chapter consists of two
parts: the fundamentals and the applications, which make it
suitable for courses of various lengths, levels, and topics based
on the mathematical background of the students. The fundamentals
portion of the chapter is intended to be read with minimal outside
assistance, and to learn some of the most useful tools in
commutative algebra. The applications of the chapter are to provide
a glimpse of the advanced mathematical research where the topics
and results are related to the material presented earlier. In the
applications portion, the authors present a number of results from
a wide range of sources without detailed proofs. The applications
portion of the chapter is suitable for a reader who knows a little
commutative algebra and algebraic geometry already, and serves as a
guide to some interesting research topics. This book should be
thought of as an introduction to more advanced texts and research
topics. Its novelty is that the material presented is a unique
combination of the essential methods and the current research
results. The goal is to equip readers with the fundamental
classical algebra and geometry tools, ignite their research
interests, and initiate some potential research projects in the
related areas.
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