© 2016

Computational Linear and Commutative Algebra


Table of contents

  1. Front Matter
    Pages I-XVIII
  2. Martin Kreuzer, Lorenzo Robbiano
    Pages 1-45
  3. Martin Kreuzer, Lorenzo Robbiano
    Pages 47-93
  4. Martin Kreuzer, Lorenzo Robbiano
    Pages 95-129
  5. Martin Kreuzer, Lorenzo Robbiano
    Pages 131-184
  6. Martin Kreuzer, Lorenzo Robbiano
    Pages 185-242
  7. Martin Kreuzer, Lorenzo Robbiano
    Pages 243-309
  8. Back Matter
    Pages 311-321

About this book


This book combines, in a novel and general way, an extensive development of the theory of families of commuting matrices with applications to zero-dimensional commutative rings, primary decompositions and polynomial system solving. It integrates the Linear Algebra of the Third Millennium, developed exclusively here, with classical algorithmic and algebraic techniques. Even the experienced reader will be pleasantly surprised to discover new and unexpected aspects in a variety of subjects including eigenvalues and eigenspaces of linear maps, joint eigenspaces of commuting families of endomorphisms, multiplication maps of zero-dimensional affine algebras, computation of primary decompositions and maximal ideals, and solution of polynomial systems.

This book completes a trilogy initiated by the uncharacteristically witty books Computational Commutative Algebra 1 and 2 by the same authors. The material treated here is not available in book form, and much of it is not available at all. The authors continue to present it in their lively and humorous style, interspersing core content with funny quotations and tongue-in-cheek explanations.


commuting endomorphisms generalized eigenspace multiplication map zero-dimensional affine algebra primary decomposition polynomial system

Authors and affiliations

  1. 1.Fakultät für Informatik und MathematikUniversität PassauPassauGermany
  2. 2.Department of MathematicsUniversity of GenoaGenoaItaly

About the authors

Martin Kreuzer holds the Chair of Symbolic Computation at the University of Passau, Germany. Starting out in Commutative Algebra and Algebraic Geometry, his research interests have developed further into Computer Algebra and its applications, including industrial applications and algebraic cryptography. He is the author or co-author of five monographs on computational algebra, cryptography and logic.

In his spare time, he plays correspondence chess, for which he is an international grandmaster and a severalfold world team champion.

Lorenzo Robbiano is a retired professor at the University of Genova, Italy. He is the co-author (with Martin Kreuzer) of the two books “Computational Commutative Algebra 1” and “Computational Commutative Algebra 2”.

Since 1987 he has been the team leader of the project CoCoA. His research interests have evolved from Algebraic Geometry to Commutative Algebra, and in the last years to Computer Algebra.

Bibliographic information


“The monograph could be used as a complementary source for classical Linear Algebra as well as an introductory book to Commutative Algebra and a starting lecture for Computer Algebra. For an interested reader it could be also a research monograph for an introduction to modern algebra. Even an experienced reader will discover new and unexpected aspects of the theory.” (Peter Schenzel, zbMATH 1360.13001, 2017)

“The book is well-written and includes many examples. Each chapter begins with a summary that motivates the … mathematics to follow, and every method is accompanied by an algorithms … . The book contains many new results and concepts, along with known ideas drawn from a widely scattered literature. … Overall, this book is a worthy contribution to both linear and commutative algebra.” (David A. Cox, Computeralgebra Rundbrief, 2017)

“The book is a textbook for advanced undergraduate and for graduate courses. Surprisingly, the experienced reader will also find new and unexpected aspects. I like the humorous style of the authors. The funny quotations help one enjoy the topic.” (Mathematical Reviews, 2017)