Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 1254.11116
Belabas, Karim; van Hoeij, Mark; Klüners, Jürgen; Steel, Allan
Factoring polynomials over global fields. (English)
[J] J. Théor. Nombres Bordx. 21, No. 1, 15-39 (2009). ISSN 1246-7405

The authors improve on {\it M. van Hoeij}'s earlier algorithm for factoring polynomials [J. Number Theory 95, 167--189 (2002; Zbl 1081.11080)]. The new algorithm is shown to run in polynomial time for polynomials over the rational numbers $\mathbb{Q}$ and over rational function fields with finite constant field ${\mathbb{F}}_{q}$. One essential idea is to work with logarithmic derivatives of polynomials $g(X)$ rather than with power sums of their zeros. (We recall that the $i$-th trace $\text{Tr}_i(g)$ is the sum of the $i$-th powers of the zeros of $g$ and that $g'/g$ equals the sum of $\text{Tr}_i(g)X^{-i-1}$ over all non-negative indices $i$.) This simplifies complexity proofs and also has practical algorithmic advantages. Following the introduction and some notations the authors give a survey on their algorithm for global fields $K$. In the final two sections they consider the special cases $K={\mathbb{F}}_{q} (t)$ and $K=\mathbb{Q}$. Descriptions of the corresponding factoring algorithms and explicit complexity proofs are given. Also, ideas for variants/improvements and -- in case $K=\mathbb{Q}$ -- comparisons with earlier algorithms are presented in some detail.
[Michael Pohst (Düsseldorf)]

MSC 2000:: *11Y40 Algebraic number theory computations
11Y16 Algorithms
11R09 Polynomials over global fields

Keywords: global fields; factoring polynomials

Citations: Zbl 1081.11080

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Article Journal Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]