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Computer Science - Theoretical Computer Science | Effective Polynomial Computation

Effective Polynomial Computation

Zippel, Richard

1993, XI, 363 p.

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  • About this book

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.

Content Level » Research

Keywords » Approximation - Diophantine approximation - Interpolation - Mathematica - algebra - algorithms - computer - computer algebra - finite field - number theory

Related subjects » Algebra - Number Theory and Discrete Mathematics - Theoretical Computer Science

Table of contents 

1. Euclid's Algorithm. 2. Continued Fractions. 3. Diophantine Equations. 4. Lattice Techniques. 5. Arithmetic Functions. 6. Residue Rings. 7. Polynomial Arithmetic. 8. Polynomial GCD's: Classical Algorithms. 9. Polynomial Elimination. 10. Formal Power Series. 11. Bounds on Polynomials. 12. Zero Equivalence Testing. 13. Univariate Interpolation. 14. Multivariate Interpolation. 15. Polynomial GCD's: Interpolation Algorithms. 16. Hensel Algorithms. 17. Sparse Hensel Algorithms. 18. Factoring over Finite Fields. 19. Irreducibility of Polynomials. 20. Univariate Factorization. 21. Multivariate Factorization. List of Symbols. Bibliography. Index.

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