Kaiserslautern - Fachbereich Mathematik
Refine
Year of publication
- 2013 (1) (remove)
Document Type
- Doctoral Thesis (1) (remove)
Language
- English (1)
Has Fulltext
- yes (1)
Faculty / Organisational entity
Factorization of multivariate polynomials is a cornerstone of many applications in computer algebra. To compute it, one uses an algorithm by Zassenhaus who used it in 1969 to factorize univariate polynomials over \(\mathbb{Z}\). Later Musser generalized it to the multivariate case. Subsequently, the algorithm was refined and improved.
In this work every step of the algorithm is described as well as the problems that arise in these steps.
In doing so, we restrict to the coefficient domains \(\mathbb{F}_{q}\), \(\mathbb{Z}\), and \(\mathbb{Q}(\alpha)\) while focussing on a fast implementation. The author has implemented almost all algorithms mentioned in this work in the C++ library factory which is part of the computer algebra system Singular.
Besides, a new bound on the coefficients of a factor of a multivariate polynomial over \(\mathbb{Q}(\alpha)\) is proven which does not require \(\alpha\) to be an algebraic integer. This bound is used to compute Hensel lifting and recombination of factors in a modular fashion. Furthermore, several sub-steps are improved.
Finally, an overview on the capability of the implementation is given which includes benchmark examples as well as random generated input which is supposed to give an impression of the average performance.