Algorithmic Algebra

1 Introduction.- 1.1 Prologue: Algebra and Algorithms.- 1.2 Motivations.- 1.2.1 Constructive Algebra.- 1.2.2 Algorithmic and Computational Algebra.- 1.2.3 Symbolic Computation.- 1.2.4 Applications.- 1.3 Algorithmic Notations.- 1.3.1 Data Structures.- 1.3.2 Control Structures.- 1.4 Epilogue.- Bibliographic Notes.- 2 Algebraic Preliminaries.- 2.1 Introduction to Rings and Ideals.- 2.1.1 Rings and Ideals.- 2.1.2 Homomorphism, Contraction and Extension.- 2.1.3 Ideal Operations.- 2.2 Polynomial Rings.- 2.2.1 Dickson’s Lemma.- 2.2.2 Admissible Orderings on Power Products.- 2.3 Gröbner Bases.- 2.3.1 Gröbner Bases in K[x1, x2,...,xn].- 2.3.2 Hilbert’s Basis Theorem.- 2.3.3 Finite 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.- 3.2.1 Example: Computable Field.- 3.2.2 Example: Ring of Integers.- 3.3 Head Reductions and Gröbner Bases.- 3.3.1 Algorithm to Compute Head Reduction.- 3.3.2 Algorithm to Compute Gröbner Bases.- 3.4 Detachability Computation.- 3.4.1 Expressing with the Gröbner Basis.- 3.4.2 Detachability.- 3.5 Syzygy Computation.- 3.5.1 Syzygy of a Gröbner Basis: Special Case.- 3.5.2 Syzygy of a Set: General Case.- 3.6 Hilbert’s Basis Theorem: Revisited.- 3.7 Applications of Gröbner Bases Algorithms.- 3.7.1 Membership.- 3.7.2 Congruence, Subideal and Ideal Equality.- 3.7.3 Sum and Product.- 3.7.4 Intersection.- 3.7.5 Quotient.- 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.- 4.3.1 Dimension of an Ideal.- 4.3.2 Solvability: Hilbert’s Nullstellensatz.- 4.3.3 Finite Solvability.- 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.- 7.3.1 Evaluation Homomorphism.- 7.4 Repeated Factors in Polynomials and Discriminants.- 7.5 Determinant Polynomial.- 7.5.1 Pseudodivision: Revisited.- 7.5.2 Homomorphism and Pseudoremainder.- 7.6 Polynomial Remainder Sequences.- 7.7 Subresultants.- 7.7.1 Subresultants and Common Divisors.- 7.8 Homomorphisms and Subresultants.- 7.9 Subresultant Chain.- 7.10 Subresultant Chain Theorem.- 7.10.1 Habicht’s Theorem.- 7.10.2 Evaluation Homomorphisms.- 7.10.3 Subresultant Chain Theorem.- 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.- 8.5.1 Real Algebraic Number Field.- 8.5.2 Root Separation, Thorn’s Lemma and Representation.- 8.6 Real Geometry.- 8.6.1 Real Algebraic Sets.- 8.6.2 Delineability.- 8.6.3 Tarski-Seidenberg Theorem.- 8.6.4 Representation and Decomposition of Semialgebraic Sets.- 8.6.5 Cylindrical Algebraic Decomposition.- 8.6.6 Tarski Geometry.- Problems.- Solutions to Selected Problems.- Bibliographic Notes.- Appendix A: Matrix Algebra.- A.1 Matrices.- A.2 Determinant.- A.3 Linear Equations.