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Table of contents
Introduction
1. Motivation
2. State of the art
3. Contributions
4. Perspectives
I. Preliminaries
1. Algebra and geometry
1.1. Commutative algebra
1.1.1. Ideals
1.1.2. Graded rings
1.1.3. Polynomial algebras
1.1.4. Combinatorics of monomials
1.1.5. Hilbert series
1.2. Regularity properties
1.2.1. Regular sequences
1.2.2. Description of the Hilbert series of a homogeneous regular sequence
1.2.3. Noether position
1.2.4. Semi-regular sequences
1.3. Algebraic geometry
1.3.1. Nullstellensatz
1.3.2. Zariski topology
1.3.3. Morphisms and rational maps
1.3.4. Dimension and degree
1.3.5. Geometric interpretation of the regularity properties
1.3.6. Singularities
1.3.7. Real semi-algebraic geometry
1.4. Genericity
1.4.1. Defnition
1.4.2. Generic properties of homogeneous systems
1.4.3. Generic changes of coordinates
1.5. Determinantal varieties
1.5.1. Defnition
1.5.2. Cramer’s formula and Schur’s complement
1.5.3. Incidence varieties
2. Gröbner bases
2.1. Monomial orderings
2.1.1. Defnition
2.1.2. Lexicographical ordering
2.1.3. Graded reverse-lexicographical ordering
2.1.4. Elimination orders
2.2. Gröbner bases: defnition
2.3. Applications
2.3.1. Zero-dimensional systems
2.3.2. Positive-dimensional systems and eliminations
2.4. Algorithms
2.4.1. A pairs algorithm: Buchberger’s algorithm
2.4.2. A matrix algorithm: Matrix-F5
2.4.3. Matrix algorithms in the axne cas
2.4.4. Change of order: FGLM
2.5. Complexity results
2.5.1. Complexity model and notations
2.5.2. Complexity of Matrix-F5 and degree of regularity
2.5.3. Thin-grained complexity of Matrix-F5
2.5.4. Complexity of FGLM
II. Contributions
3. Weighted homogeneous systems
3.1. Introduction
3.2. Properties
3.2.1. Defnitions and properties
3.2.2. Degree and Bézout’s bound
3.2.3. Changes of variables and reverse chain-divisible systems of weights
3.2.4. Genericity of regularity properties andW-compatibility
3.2.5. Characterization of the Hilbert series
3.3. W-Degree of regularity of regular sequences
3.3.1. Macaulay’s bound in dimension zero
3.3.2. Sharper bound on the degree of regularity
3.3.3. Conjectured exact formula
3.3.4. Positive-dimensional regular sequences
3.4. Overdetermined systems
3.4.1. Denition of semi-regularity
3.4.2. Characterization with the Hilbert series
3.4.3. W-degree of regularity of semi-regular sequences
3.4.4. Fröberg’s conjecture
3.5. Algorithms and complexity
3.5.1. Critical pairs algorithms
3.5.2. Adapting the Matrix-F5 algorithm
3.5.3. Thin-grained complexity of Matrix-F5
3.5.4. FGLM and computational strategy for zero-dimensional systems
3.5.5. Other algorithms
3.6. Experiments
3.6.1. Generic systems
3.6.2. Discrete Logarithm Problem
3.6.3. Polynomial inversion
4. Real roots classification for determinants – Applic. to contrast optimization
4.1. Introduction
4.2. Modeling the dynamics of the contrast optimization
4.3. Algorithm
4.3.1. Classifcation strategy
4.3.2. The determinantal problem
4.3.3. Incidence varieties
4.3.4. Locus of rank exactly r0
4.3.5. Singularities
4.3.6. Boundary
4.4. The contrast problem
4.4.1. The case of water
4.4.2. The general case
III. Appendices
A. Source code
A.1. Magma source code for Matrix-F5
A.1.1. Algorithm Matrix-F5
A.1.2. Weighted homogeneous systems
A.2. Maple source code for determinantal varieties (Chapter 4)
A.2.1. Algorithms
A.2.2. Case of water
A.2.3. General case
