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| Introduction 1 Constructive Ideal Theory 1.1 Ideals and GrSbner Bases 1.2 Elimination Ideals 1.3 Syzygy Modules 1.4 Hilbert Series 1.5 The Radical Ideal 1.6 Normalization 2 Invariant Theory 2.1 Invariant Rings 2.2 Reductive Groups 2.3 Categorical Quotients 2.4 Homogeneous Systems of Parameters 2.5 The Cohen-Macaulay Property of Invariant Rings 2.6 Hilbert Series of Invariant Rings 3 Invariant Theory of Finite Groups 3.1 Homogeneous Components 3.2 Moliens Formula.. 3.3 Primary Invariants 3.4 Cohen-Macaulayness 3.5 Secondary Invariants 3.6 Minimal Algebra Generators and Syzygies 3.7 Properties of Invariant Rings 3.8 Noethers Degree Bound 3.9 Degree Bounds in the Modular Case 3.10 Permutation Groups 3.11 Ad Hoc Methods 4 Invariant Theory of Reductive Groups 4.1 Computing Invariants of Linearly Reductive Groups 4.2 Improvements and Generalizations 4.3 Invariants of Tori 4.4 Invariants of SLn and GLn 4.5 The Reynolds Operator 4.6 Computing Hilbert Series 4.7 Degree Bounds for Invariants 4.8 Properties of Invariant Rings 5 Applications of Invariant Theory 5.1 Cohomology of Finite Groups 5.2 Galois Group Computation 5.3 Noethers Problem and Generic Polynomials 5.4 Systems of Algebraic Equations with Symmetries 5.5 Graph Theory 5.6 Combinatorics 5.7 Coding Theory 5.8 Equivariant Dynamical Systems 5.9 Material Science 5.10 Computer Vision A Linear Algebraic Groups A.1 Linear Algebraic Groups A.2 The Lie Algebra of a Linear Algebraic Group A.3 Reductive and Semi-simple Groups A.4 Roots A.5 Representation Theory References Notation Index |
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