Factoring univariate polynomials over the rationals

The complexity of factoring univariate polynomials over the rationals

This tutorial will explain the algorithm behind the currently fastest implementations for univariate factorization over the rationals. The complexity will be analyzed; it turns out that modifications were needed in order to prove a polynomial time complexity while preserving the best practical performance. The complexity analysis leads to two results: (1) it shows that the practical performance...

Complexity results for factoring univariate polynomials over the rationals (version 0.3)

In [6] Zassenhaus gave an algorithm for factoring polynomials f ∈ Q[x]. In this algorithm one has to solve a combinatorial problem of size r, where r is the number of local factors of f at some suitably chosen prime p. This combinatorial problem consists of selecting the right subsets of the set of local factors. In the worst case, the algorithm [6] ends up trying 2r−1 such subsets (if a subset...

Improved Techniques for Factoring Univariate Polynomials

The paper describes improved techniques for factoring univariate polynomials over the integers. The authors modify the usual linear method for lifting modular polynomial factorizations so that efficient early factor detection can be performed. The new lifting method is universally faster than the classical quadratic method, and is faster than a linear method due to Wang, provided we lift suffic...

Factoring Polynomials over Local Fields

Factoring Polynomials over Number Fields

The purpose of these notes is to give a substantially self-contained introduction to the factorization of polynomials over number fields. In particular, we present Zassenhaus’ algorithm and a factoring algorithm using lattice reduction, which were, respectively, the best in practice and in theory, before 2002. We give references for the van Hoeij-Novocin algorithm, currently the best both in pr...