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Table of contents
Introduction
I Preliminaries
1 Gröbner Bases
1.1 Gröbner Basics
1.1.1 Ideals and Varieties
1.1.2 Monomial Orderings and Gröbner bases
1.1.3 Buchberger Algorithm
1.1.4 What is Solving ?
1.2 Gröbner bases and Linear Algebra
1.2.1 Lazard’s algorithm and Macaulay’s matrices
1.2.2 Matrix-F5 algorithm
1.2.3 FGLM algorithm
1.3 Extension to subalgebras
1.3.1 SAGBI bases
1.3.2 Matrix SAGBI-F5 algorithm
2 Commutative Algebra and Gröbner bases
2.1 Commutative Algebra and Hilbert series
2.1.1 Algebraic tools
2.1.2 Gradings on subalgebras of K[X±1]
2.1.3 Hilbert Function and Hilbert Series
2.2 Applications in K[X]
2.2.1 Bounds on the degrees
2.2.2 Genericity of regular sequences. Semi-regular sequences
2.2.3 Affine case
3 Invariant Theory and Monomial Algebras
3.1 Invariant Theory
3.1.1 Action of Groups on Polynomials. Computation of Invariants
3.1.2 Molien’s Theorem
3.1.3 Structure of the algebra of invariants, and classical strategies
3.1.4 Representation Theory of finite groups
3.1.5 Estimates of Dimensions of Isotypic Components
3.2 Monomial Algebras
II Contributions
4 Solving systems with symmetries
4.1 Vortex Problem
4.1.1 Vortex Problem
4.1.2 From invariant system to invariant equations
4.1.3 From two blocks to symmetric functions in one block
4.1.4 Solving the equations with the symmetric functions
4.1.5 Benchmarks
4.2 Ideals stable under the action of an abelian group
4.2.1 Linear change of variables
4.2.2 Grading induced by a diagonal matrix group
4.2.3 Abelian Matrix-F5 algorithm
4.2.4 Abelian-FGLM algorithm
4.2.5 Complexity questions
4.2.6 Experiments
4.3 Stable equations and SAGBI bases
4.3.1 SAGBI-Gröbner bases in invariant rings
4.3.2 SAGBI-FGLM algorithm and general algorithm to obtain an invariant Gröbner basis
4.3.3 Removing spurious solutions
4.3.4 Implementation and Benchmarks
5 Gröbner Bases in Monomial Algebras
5.1 Introduction
5.2 Sparse Gröbner bases
5.3 Algorithms
5.3.1 Sparse-MatrixF5 algorithm
5.3.2 Sparse-FGLM algorithm
5.4 Complexity
5.5 Dense, multi-homogeneous and overdetermined systems
5.6 Experimental results
