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Preface

Published online by Cambridge University Press:  05 June 2013

Teo Mora
Affiliation:
University of Genoa
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Summary

If you HOPE that this second SPES volume preserves the style of the previous volume, you will not be disappointed: in fact it maintains a self-contained approach using only undergraduate mathematics in this introduction to elementary commutative ideal theory and to its computational aspects, while my horror vacui compelled me to report nearly all the relevant results in computational algebraic geometry that I know about.

When the commutative algebra community was exposed, in 1979, to Buchberger's theory and algorithm (dated 1965) of Gröbner bases, the more alert researchers, mainly Schreyer and Bayer, immediately realized that this injection of Gröbner technology was all one needed to make effective Macaulay's paradigm for reducing computational problems for ideals either to the corresponding combinatorial problem for monomials or to a more elementary linear algebraic computation.4 This realization gave to researchers a straightforward approach which led them, within more or less fifteen years, to completely effectivize commutative ideal theory.

This second volume of SPES is an eyewitness report on this successful introduction of effective methods to algebraic geometry.

Part three, Gauss, Euclid, Buchberger: Elementary Gröbner Bases, introduces at the same time Buchberger's theory of Gröbner bases, his algorithm for computing them and Macaulay's paradigm.

While I will discuss in depth both of the classical main approaches to the introduction of Gröbner bases – their relation with rewriting rules and the Knuth–Bendix Algorithm, and their connection with Macaulay's H-bases and Hironaka's standard bases as tools for lifting properities to a polynomial algebra from its graded algebra – my presentation stresses the relation of both the notion and the algorithm to elementary linear algebra and Gaussian reduction; an added bonus of this approach is the ability to link Buchberger's algorithm with the most recent alternative linear algebra approach proposed by Faugère.

Type
Chapter
Information
Solving Polynomial Equation Systems II
Macaulay's Paradigm and Gröbner Technology
, pp. xi - xiii
Publisher: Cambridge University Press
Print publication year: 2005

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  • Preface
  • Teo Mora, University of Genoa
  • Book: Solving Polynomial Equation Systems II
  • Online publication: 05 June 2013
  • Chapter DOI: https://doi.org/10.1017/CBO9781107340954.001
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  • Preface
  • Teo Mora, University of Genoa
  • Book: Solving Polynomial Equation Systems II
  • Online publication: 05 June 2013
  • Chapter DOI: https://doi.org/10.1017/CBO9781107340954.001
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Preface
  • Teo Mora, University of Genoa
  • Book: Solving Polynomial Equation Systems II
  • Online publication: 05 June 2013
  • Chapter DOI: https://doi.org/10.1017/CBO9781107340954.001
Available formats
×