As indicated in the Preface, solutions of the fundamental questions addressed in this book, the center and cyclicity problems, are expressed in terms of the sets of common zeros of collections of polynomials in the coefficients of the underlying family of systems of differential equations. These sets of common zeros are termed varieties. They are determined not so much by the specific polynomials themselves as by larger collections of polynomials, the so-called ideals that the original collections of polynomials generate. In the first section of this chapter we discuss these basic concepts: polynomials, varieties, and ideals. An ideal can have more than one set of generating polynomials, and a fundamental problem is that of deciding when two ideals, hence the varieties they determine, are the same, even though presented by different sets of generators. To address this and related isssues, in Sections 1.2 and 1.3 we introduce the concept of a Gröobner basis and certain fundamental techniques and algorithms of computational algebra for the study of polynomial ideals and their varieties. The last section is devoted to the decomposition of varieties into their simplest components and shows how this decomposition is connected to the structure of the generating ideals. For a fuller exposition of the concepts presented here, the reader may consult [1, 18, 23, 60].
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2009 Birkhäuser Boston
About this chapter
Cite this chapter
Romanovski, V., Shafer, D. (2009). Polynomial Ideals and Their Varieties. In: The Center and Cyclicity Problems. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4727-8_1
Download citation
DOI: https://doi.org/10.1007/978-0-8176-4727-8_1
Published:
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-4726-1
Online ISBN: 978-0-8176-4727-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)