Univariate polynomial equation solving

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Table of contents

Introduction
I Background
Chapter 1 Gröbner bases
1.1 Preliminaries
1.2 Gröbner bases and Buchberger algorithm
1.3 FGLM algorithm
1.4 Gröbner bases for polynomial system solving
Chapter 2 Triangular sets
2.1 Concepts and terminologies
2.2 Characteristic sets and Wu–Ritt algorithm
2.3 Regular, simple and irreducible triangular sets
2.4 Triangular sets for polynomial system solving
Chapter 3 Some constructions in algebra
3.1 Commutative algebra
3.2 Basics of finite fields
Chapter 4 Sparse FGLM algorithms
4.1 Ideals in shape position
4.2 General ideals
4.3 Multiplication matrices
4.4 Implementation and experimental results
Chapter 5 Simple decomposition over finite fields
5.1 Simple sets revisited
5.2 Zero-dimensional polynomial sets
5.3 Positive-dimensional polynomial sets
5.4 Implementation and experimental results
Chapter 6 Squarefree decomposition and factorization over unmixed products of field extensions
6.1 Unmixed products of field extensions
6.2 Squarefree decomposition over unmixed products
6.3 Factorization over unmixed products
6.4 Examples and experiments
Chapter 7 Applications
7.1 Detection of steady states and their numbers for finite biological models
7.2 Sparse FGLM algorithm for interpolation problem in list decoding
Conclusions
Bibliography