Fundamental Problems of Algorithmic Algebra

Popular computer algebra systems such as Maple, Macsyma, Mathematica, and REDUCE are now basic tools on most computers. Efficient algorithms for various algebraic operations underlie all these systems. Computer algebra, or algorithmic algebra, studies these algorithms and their properties and represents a rich intersection of theoretical computer science with classical mathematics. Fundamental Problems of Algorithmic Algebra provides a systematic and focused treatment of a collection of core problemsthe computational equivalents of the classical Fundamental Problem of Algebra and its derivatives. Topics covered include the GCD, subresultants, modular techniques, the fundamental theorem of algebra, roots of polynomials, Sturm theory, Gaussian lattice reduction, lattices and polynomial factorization, linear systems, elimination theory, Grobner bases, and more. Features · Presents algorithmic ideas in pseudo-code based on mathematical concepts and can be used with any computer mathematics system · Emphasizes the algorithmic aspects of problems without sacrificing mathematical rigor · Aims to be self-contained in its mathematical development · Ideal for a first course in algorithmic or computer algebra for advanced undergraduates or beginning graduate students

Table of Contents:
O INTRODUCTION 1: Fundamental Problem of Algebra 2: Fundamental Problem of Classical Algebraic Geometry 3: Fundamental Problem of Ideal Theory 4: Representation and Size 5: Computational Models 6: Asymptotic Notations 7: Complexity of Multiplication 8: On Bit versus Algebraic Complexity 9: Miscellany 10: Computer Algebra Systems I ARITHMETIC 1: The Discrete Fourier Transform 2: Polynomial Multiplication 3: Modular FFT 4: Fast Integer Multiplication 5: Matrix Multiplication II THE GCD 1: Unique Factorization Domain 2: Euclid's Algorithm 3: Euclidean Ring 4: The Half-GCD problem 5: Properties of the Norm 6: Polynomial HGCD A: APPENDIX: Integer HGCD III SUBRESULTANTS 1: Primitive Factorization 2: Pseudo-remainders and PRS 3: Determinantal Polynomials 4: Polynomial Pseudo-Quotient 5: The Subresultant PRS 6: Subresultants 7: Pseudo-subresultants 8: Subresultant Theorem 9: Correctness of the Subresultant PRS Algorithm IV MODULAR TECHNIQUES 1: Chinese Remainder Theorem 2: Evaluation and Interpolation 3: Finiding Prime Moduli 4: Lucky homomorphisms for the GCD 5: Coefficient Bounds for Factors 6: A Modular GCD algorithm 7: What else in GCD computation? IV FUNDAMENTAL THEOREM OF ALGEBRA 1: Elements of Field Theory 2: Ordered Rings 3: Formally Real Rings 4: Constructible Extensions 5: Real Closed Fields 6: Fundamental Theorem of Algebra VI ROOTS OF POLYNOMIALS 1: Elementary Properties of Polynomial Roots 2: Root Bounds 3: Algebraic Numbers 4: Resultants 5: Symmetric Functions 6: Discriminant 7: Root Separation 8: A Generalized Hadamard Bound 9: Isolating Intervals 10: On Newton's Method 11: Guaranteed Convergence of Newton Iteration VII STURM THEORY 1: Sturm Sequences from PRS 2: A Generalized Sturm Theorem 3: Corollaries and Applications 4: Integer and Complex Roots 5: The Routh-Hurwitz Theorem 6: Sign Encoding of Algebraic Numbers: Thom's Lemma 7: Problem of Relative Sign Conditions 8: The BKR algorithm VIII GAUSSIAN LATTICE REDUCTION 1: Lattices 2: Shortest vectors in planar lattices 3: Coherent Remainder Sequences IX LATTICE REDUCTION AND APPLICATIONS 1: Gram-Schmidt Orthogonalization 2: Minkowski's Convex Body Theorem 3: Weakly Reduced Bases 4: Reduced Bases and the LLL-algorithm 5: Short Vectors 6: Factorization via Reconstruction of Minimal Polynomials X LINEAR SYSTEMS 1: Sylvester's Identity 2: Fraction-free Determinant Computation 3: Matrix Inversion 4: Hermite Normal Form 5: A Multiple GCD Bound and Algorithm 6: Hermite Reduction Step 7: Bachem-Kannan Algorithm 8: Smith Normal Form 9: Further Applications XI ELIMINATION THEORY 1: Hilbert Basis Theorem 2: Hilbert Nullstellensatz 3: Specializations 4: Resultant Systems 5: Sylvester Resultant Revisited 6: Interial Ideal 7: The Macaulay Resultant 8: U-Resultant 9: Generalized Characteristic Polynomial 10: Generalized U - resultant 11: A Multivariate Root Bound A: APPENDIX A: Power Series B: APPENDIX B: Counting Irreducible Polynomials XII GROBNER BASES 1: Admissible Orderings 2: Normal Form ALgorithm 3: Characterizations of Grobner Bases 4: Buchberger's Algorithm 5: Uniqueness 6: Elimination Properties 7: Computing in Quotient Rings 8: Syzygies XIII BOUNDS IN POLYNOMIAL IDEAL THEORY 1: Some Bounds in Polynomial Ideal Theory 2: The Hilbert-Sette Theorem 3: Homogeneous Sets 4: Cone Decomposition 5: Exact Decomposition of NF (I) 6: Exact Decomposition of Ideals 7: Bouding the Macaulay constants 8: Term Rewriting Systems 9: A Quadratic Counter 10: Uniqueness Property 11: Lower Bounds A: APPENDIX: Properties of So XIV CONTINUED FRACTIONS 1: Introductions 2: Extended Numbers 3: General Terminology 4: Ordinary Continued Fractions 5: Continued fractions as Mobius transformations 6: Convergence Properties 7: Real Mobius Transformations 8: Continued Fractions of Roots 9: Arithmetic Operations