Algorithmic Algebra

This book is based on a graduate computer science course entitled "Symbolic Computational Algebra" taught by the author at New York University. This book is meant for graduate students with training in theoretical computer science, who would like to do research in computational algebra or to understand the algorithms underlying currently available symbolic computational systems such as Mathematica, Maple or Axiom. The four main topics covered are Gröbner bases, characteristic sets, resultants and semialgebraic sets.

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.