# Unique factorization domain package.

This package contains classes for polynomials rings as unique factorization domains. Provided methods with interface `GreatestCommonDivisor` are e.g. greatest common divisors `gcd()`, primitive part `primitivePart()` or `coPrime()`. The different classes implement variants of polynomial remainder sequences (PRS) and modular methods. Interface `Squarefree` provides the greatest squarefree factor `squarefreeFactor()` and a complete squarefree decompostion can be obtained with method `squarefreeFactors()`. There is a `Factorization` interface with an `FactorAbstract` class with common codes. Factorization of univariate polynomials exists for several coefficient rings: modulo primes in class `FactorModular`, over integers in class `FactorInteger`, over rational numbers in class `FactorRational`, over algebraic numbers in class `FactorAlgebraic<C>` and over rational functions in class `FactorQuotient<C>` (where for the last two classes `C` can be any other ring for which the `FactorFactory. getImplementation` returns an implementation). Multivatiate polynomials are reduced to the univariate polynomials via Kronecker substitution and are therefore not very efficient at the moment. The factorization of polynomials is partly experimental. The rational function class `Quotient` computes quotients of polynomials reduced to lowest terms.

To choose the correct implementation always use the factory classes `GCDFactory`, `SquarefreeFactory` and `FactorFactory` with methods `getImplementation()` or `getProxy()`. These methods will take care of all possible (implemented) coefficient rings properties. The polynomial coefficients must implement the `GcdRingElem` interface and so must allow greatest common divisor computations. Greatest common divisor computation is completely generic and works for any implemented integral domain. If special, optimized implementations exist they will be used. Squarefree decomposition is also completely generic and works for any implemented integral domain. There are no special, optimized implementations. Factorization is generic relative to the implemented ring constructions: algebraic field extensions and transcendent field extensions. Implemented base cases are modular coefficient, integer coefficients and rational number coefficients.

The implementation follows Geddes & Czapor & Labahn Algorithms for Computer Algebra and Cohen A Curse in Computational Algebraic Number Theory. See also Kaltofen Factorization of Polynomials in Computing Supplement, Springer, 1982, Davenport & Gianni & Trager Scratchpad's View of Algebra II: A Categorical View of Factorization in ISSAC'91 and the ALDES/SAC2 code as contained in MAS.

Last modified: Sat Oct 23 18:55:42 CEST 2010

$Id: package.html 3357 2010-10-23 17:10:02Z kredel $