Algorithmic Algebra

1 Introduction.- 1.1 Prologue: Algebra and Algorithms.- 1.2 Motivations.- 1.3 Algorithmic Notations.- 1.4 Epilogue.- Bibliographic Notes.- 2 Algebraic Preliminaries.- 2.1 Introduction to Rings and Ideals.- 2.2 Polynomial Rings.- 2.3 Gröbner Bases.- 2.4 Modules and Syzygies.- 2.5 S-Polynomials.- Problems.- Solutions to Selected Problems.- Bibliographic Notes.- 3 Computational Ideal Theory.- 3.1 Introduction.- 3.2 Strongly Computable Ring.- 3.3 Head Reductions and Gröbner Bases.- 3.4 Detachability Computation.- 3.5 Syzygy Computation.- 3.6 Hilbert’s Basis Theorem: Revisited.- 3.7 Applications of Gröbner Bases Algorithms.- Problems.- Solutions to Selected Problems.- Bibliographic Notes.- 4 Solving Systems of Polynomial Equations.- 4.1 Introduction.- 4.2 Triangular Set.- 4.3 Some Algebraic Geometry.- 4.4 Finding the Zeros.- Problems.- Solutions to Selected Problems.- Bibliographic Notes.- 5 Characteristic Sets.- 5.1 Introduction.- 5.2 Pseudodivision and Successive Pseudodivision.- 5.3 Characteristic Sets.- 5.4 Properties of Characteristic Sets.- 5.5 Wu-Ritt Process.- 5.6 Computation.- 5.7 Geometric Theorem Proving.- Problems.- Solutions to Selected Problems.- Bibliographic Notes.- 6 An Algebraic Interlude.- 6.1 Introduction.- 6.2 Unique Factorization Domain.- 6.3 Principal Ideal Domain.- 6.4 Euclidean Domain.- 6.5 Gauss Lemma.- 6.6 Strongly Computable Euclidean Domains.- Problems.- Solutions to Selected Problems.- Bibliographic Notes.- 7 Resultants and Subresultants.- 7.1 Introduction.- 7.2 Resultants.- 7.3 Homomorphisms and Resultants.- 7.4 Repeated Factors in Polynomials and Discriminants.- 7.5 Determinant Polynomial.- 7.6 Polynomial Remainder Sequences.- 7.7 Subresultants.- 7.8 Homomorphisms and Subresultants.- 7.9 Subresultant Chain.- 7.10 Subresultant ChainTheorem.- Problems.- Solutions to Selected Problems.- Bibliographic Notes.- 8 Real Algebra.- 8.1 Introduction.- 8.2 Real Closed Fields.- 8.3 Bounds on the Roots.- 8.4 Sturm’s Theorem.- 8.5 Real Algebraic Numbers.- 8.6 Real Geometry.- Problems.- Solutions to Selected Problems.- Bibliographic Notes.- Appendix A: Matrix Algebra.- A.1 Matrices.- A.2 Determinant.- A.3 Linear Equations.