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E-raamat: Computational Linear and Commutative Algebra

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  • Formaat: PDF+DRM
  • Ilmumisaeg: 06-Sep-2016
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319436012
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Computational Linear and Commutative AlgebraSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 06-Sep-2016
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319436012
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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.

Foreword.- Introduction.- 1 Endomorphisms.- 2 Families of Commuting Endomorphisms.- 3 Special Families of Endomorphisms.- 4 Zero-Dimensional Affine Algebras.- 5 Computing Primary and Maximal Components.- 6 Solving Zero-Dimensional Polynomial Systems.- Notation.- References.- Index.

Arvustused

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 textbookfor 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)

1 Endomorphisms
1(46)
1.1 Generalized Eigenspaces
2(10)
1.1.A Big Kernels and Small Images
3(3)
1.1.B Minimal Polynomials and Eigenspaces
6(6)
1.2 Minimal and Characteristic Polynomials
12(5)
1.3 Nilpotent Endomorphisms and Multiplicities
17(7)
1.4 The Module Structure Given by an Endomorphism
24(10)
1.5 Commendable Endomorphisms
34(6)
1.6 Other Special Endomorphisms
40(7)
2 Families of Commuting Endomorphisms
47(48)
2.1 Commuting Families
49(5)
2.2 Kernels and Big Kernels of Ideals
54(10)
2.2.A Properties of Zero-Dimensional Rings
55(4)
2.2.B Properties of Kernels and Big Kernels of Ideals
59(5)
2.3 Eigenfactors
64(5)
2.4 Joint Eigenspaces
69(8)
2.5 Splitting Endomorphisms
77(12)
2.6 Simultaneous Diagonalization and Triangularization
89(6)
3 Special Families of Endomorphisms
95(36)
3.1 F-Cyclic Vector Spaces
96(5)
3.2 Unigenerated Families
101(9)
3.3 Commendable Families
110(3)
3.4 Local Families
113(6)
3.5 Dual Families
119(7)
3.6 Extended Families
126(5)
4 Zero-Dimensional Affine Algebras
131(54)
4.1 Multiplication Endomorphisms
134(5)
4.2 Primary Decomposition and Separators
139(6)
4.3 Commendable and Splitting Multiplication Endomorphisms
145(8)
4.4 Local Multiplication Families
153(8)
4.5 Dual Multiplication Families
161(14)
4.6 Hilbert Functions and the Cayley-Bacharach Property
175(10)
5 Computing Primary and Maximal Components
185(58)
5.1 Computing Primary Decompositions
188(15)
5.1.A Using the Generically Extended Linear Form
189(5)
5.1.B Using Linear Forms and Idempotents
194(6)
5.1.C Computing Joint Eigenspaces
200(3)
5.2 Primary Decomposition over Finite Fields
203(10)
5.3 Computing Maximal Components via Factorization
213(13)
5.3.A Minimal Polynomials and Finite Field Extensions
213(5)
5.3.B Factorizing over Extension Fields
218(5)
5.3.C Using Factorizations to Compute Maximal Components
223(3)
5.4 Primary Decompositions Using Radical Ideals
226(11)
5.4.A Maximal Components via Radical Ideals
227(6)
5.4.B Primary Components from Maximal Components
233(4)
5.5 The Separable Subalgebra
237(6)
6 Solving Zero-Dimensional Polynomial Systems
243(68)
6.1 Rational Zeros via Commuting Families
245(10)
6.1.A Computing One-Dimensional Joint Eigenspaces
246(6)
6.1.B Computing Linear Maximal Ideals
252(3)
6.2 Rational Zeros via Eigenvalues and Eigenvectors
255(14)
6.2.A The Eigenvalue Method
256(8)
6.2.B The Eigenvector Method
264(5)
6.3 Solving Polynomial Systems over Finite Fields
269(26)
6.3.A Computing Isomorphisms of Finite Fields
271(8)
6.3.B Solving over Finite Fields via Cloning
279(6)
6.3.C Solving over Finite Fields via Univariate Representations
285(5)
6.3.D Solving over Finite Fields via Recursion
290(5)
6.4 Solving Polynomial Systems over the Rationals
295(16)
6.4.A Splitting Fields in Characteristic Zero
297(5)
6.4.B Solving over the Rational Numbers via Cloning
302(9)
Notation 311(4)
References 315(2)
Index 317
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.
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