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E-raamat: Effective Polynomial Computation

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Effective Polynomial ComputationSuurem pilt
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Effective Polynomial Computation is an introduction to the algorithms of computer algebra. It discusses the basic algorithms for manipulating polynomials including factoring polynomials. These algorithms are discussed from both a theoretical and practical perspective. Those cases where theoretically optimal algorithms are inappropriate are discussed and the practical alternatives are explained. Effective Polynomial Computation provides much of the mathematical motivation of the algorithms discussed to help the reader appreciate the mathematical mechanisms underlying the algorithms, and so that the algorithms will not appear to be constructed out of whole cloth. Preparatory to the discussion of algorithms for polynomials, the first third of this book discusses related issues in elementary number theory. These results are either used in later algorithms (e.g. the discussion of lattices and Diophantine approximation), or analogs of the number theoretic algorithms are used for polynomial problems (e.g. Euclidean algorithm and p-adic numbers). Among the unique features of Effective Polynomial Computation is the detailed material on greatest common divisor and factoring algorithms for sparse multivariate polynomials. In addition, both deterministic and probabilistic algorithms for irreducibility testing of polynomials are discussed.

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Springer Book Archives
1 Euclids Algorithm.- 1.1 Euclidean Algorithm.- 1.2 Diophantine
Approximations.- 1.3 Continued Fractions.- 1.4 Diophantine Equations.- 2
Continued Fractions.- 2.1 Basics.- 2.2 Matrix Representation.- 2.3 Continuant
Representation.- 2.4 Continued Fractions of Quadratics.- 2.5 Approximation
Properties.- 2.6 Continued Fraction Arithmetic.- 3 Diophantine Equations.-
3.1 Two Variable Linear Diophantine Equations.- 3.2 General Linear
Diophantine Equations.- 3.3 Pells Equation.- 3.4 Fermats Last Theorem.- 4
Lattice Techniques.- 4.1 Lattice Fundamentals.- 4.2 Minkowski Convex Body
Theorem.- 4.3 Reduced Bases.- 4.4 Finding Numerical Relationships.- 5
Arithmetic Functions.- 5.1 Arithmetic Functions.- 5.2 Asymptotic Behavior of
Arithmetic Functions.- 5.3 Distribution of Primes.- 5.4 Bertrands
Postulate.- 6 Residue Rings.- 6.1 Basic Properties of ?/m?.- 6.2 Chinese
Remainder Theorem.- 6.3 Multiplicative Structure of ?/m?.- 6.4 Quadratic
Reciprocity.- 6.5 Algebraic Extensions of.- 6.6 p-adic Numbers.- 6.7
Cryptosystems.- 6.8 Sums of Squares.- 7 Polynomial Arithmetic.- 7.1
Generalities.- 7.2 Polynomial Addition.- 7.3 Polynomial Multiplication.- 7.4
Fast Polynomial Algorithms.- 7.5 Polynomial Exponentiation.- 7.6 Polynomial
Substitution.- 8 Polynomial GCDs: Classical Algorithms.- 8.1 Generalities.-
8.2 GCD of Several Quantities.- 8.3 Polynomial Contents.- 8.4 Coefficient
Growth.- 8.5 Pseudo-Quotients.- 8.6 Subresultant Polynomial Remainder
Sequence.- 9 Polynomial Elimination.- 9.1 Symmetric Functions.- 9.2
Polynomial Resultants.- 9.3 Subresultants.- 9.4 Elimination Examples.- 10
Formal Power Series.- 10.1 Introduction.- 10.2 Power Series Arithmetic.- 10.3
Power Series Exponentiation.- 10.4 Composition of Formal Power Series.- 10.5
Reversion of Power Series.- 11Bounds on Polynomials.- 11.1 Heights of
Polynomials.- 11.2 Uniform Coefficient Bounds.- 11.3 Weighted Coefficient
Bounds.- 11.4 Size of a Polynomials Zeroes.- 11.5 Discriminants and Zero
Separation.- 12 Zero Equivalence Testing.- 12.1 Probabilistic Techniques.-
12.2 Deterministic Results.- 12.3 Negative Results.- 13 Univariate
Interpolation.- 13.1 Vandermonde Matrices.- 13.2 Lagrange Interpolation.-
13.3 Newton Interpolation.- 13.4 Fast Fourier Transform.- 13.5 Abstract
Interpolation.- 14 Multivariate Interpolation.- 14.1 Multivariate Dénse
Interpolation.- 14.2 Probabilistic Sparse Interpolation.- 14.3 Deterministic
Sparse Interpolation with Degree Bounds.- 14.4 Deterministic Sparse
Interpolation without Degree Bounds.- 15 Polynomial GCDs: Interpolation
Algorithms.- 15.1 Heuristic GCD.- 15.2 Univariate Polynomials over ?.- 15.3
Multivariate Polynomials.- 16 Hensel Algorithms.- 16.1 m-adic Completions.-
16.2 One Dimensional Iteration.- 16.3 Multidimensional Iteration.- 16.4
Hensels Lemma.- 16.5 Generalizations of Hensels Lemma.- 16.6 Zassenhaus
Formulation of Hensels Lemma.- 17 Sparse Hensel Algorithms.- 17.1 Heuristic
Presentation.- 17.2 Formal Presentation.- 18 Factoring over Finite Fields.-
18.1 Square Free Decomposition.- 18.2 Distinct Degree Factorization.- 18.3
Finding Linear Factors.- 18.4 Cantor-Zassenhaus Algorithm.- 19 Irreducibility
of Polynomials.- 19.1 Deterministic Irreducibility Testing.- 19.2 Counting
Prime Factors.- 19.3 Hilbert Irreducibility Theorem.- 19.4 Bertinis
Theorem.- 20 Univariate Factorization.- 20.1 Reductions.- 20.2 Simple
Algorithm.- 20.3 Asymptotically Good Algorithms.- 21 Multivariate
Factorization.- 21.1 General Reductions.- 21.2 Lifting Multivariate
Factorizations.- 21.3 Leading Coefficient Determination.- 21.4Multivariate
Polynomials over Q.- 21.5 Bivariate Polynomials over Fields.- List of symbols.
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