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Slide 1

Polynomial Factorization

Olga Sergeeva

Ferien-Akademie 2004, September 19 – October 1

Slide 2

Overview

Univariate Factorization

Overview of the algoriths and the required simplifications

Factoring over finite fields

Factorization based on Hensel lifting

LLL algorith

Multivariate Factorization

Problems overview

The idea of the algorith

Analysis of correctness probability.

Slide 3

We consider factorization of polynomials over the rational integers, Z, and different approaches to this problem.

Slide 4

Univariate Factorization – algoriths

We consider factorization of polynomials over the rational integers, Z, and different approaches to this problem.

Algoriths, solving the problem for univariate polynomials:

Kronecker, interpolation algorith

Slide 5

Univariate Factorization – algoriths

We consider factorization of polynomials over the rational integers, Z, and different approaches to this problem.

Algoriths, solving the problem for univariate polynomials:

Kronecker, interpolation algorith

Algorith, which uses Hensel lifting techniques and factorization over finite fields

Slide 6

Univariate Factorization – algoriths

Algoriths, solving the problem for univariate polynomials:

Kronecker, interpolation algorith

Algorith, which uses Hensel lifting techniques and factorization over finite fields

A. K. Lenstra, H. W. Lenstra and Lovasz – polynomial time algorith using basic reduction techniques for lattices.

Slide 7

When factoring a univariate polynomial over Z, the following simplifications are effective:

removing the integer content of F(Z)

Slide 8

Univariate Factorization – simplifications

When factoring a univariate polynomial over Z, the following simplifications are effective:

removing the integer content of F(Z)

computing square free decomposition (with use of GCD computations or modular interpolation techniques).

Slide 9

- Univariate Factorization – algoriths
- Univariate Factorization – simplifications
- The last slide about finite fields
- Univariate Factorization algorith (UFA)
- Hensel techniques reminder
- Asymptotically Good Algoriths
- Lattices and factorization
- Lattices and factorization: two theorems
- Auxiliary algorith
- The main algorith
- Multivariate factorization
- Hilbert irreducibility theorem
- Bertini’s theorem

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