Jürgen Gerhard

Fast algorithms for Taylor shifts and certain difference equations

Joachim von zur Gathen Jurgen Gerhard Fachbereich 17 Mathematik-Informatik Universitat-GH Paderborn D-33095 Paderborn, Germany {gathen, jngerhar}@uni-paderborn. de http://www.uni-paderborn .de/cs/gathen .html

Arithmetic and factorization of polynomial over F 2 (extended abstract)

We describe algorithms for polynomial multiplication and polynomial factorization over the binary field IF2.and their implementation. They allow polynomials of degree up to 100,000 to be factored in about one dqy of CPU time.

Polynomial factorization over F 2

We describe algorithms for polynomial factorization over the binary field F2, and their implementation. They allow polynomials of degree up to 250 000 to be factored in about one day of CPU time, distributing the work on two processors.

Shiftless decomposition and polynomial-time rational summation

New algorithms are presented for computing the dispersion set of two polynomials over Q and for shiftless factorization. Together with a summability criterion by Abramov, these are applied to get a polynomial-time algorithm for indefinite rational summation, using a sparse representation of the output.

Factoring a binary polynomial of degree over one million

On 22 May 2000, the factorization of a pseudorandom polynomial of degree 1 048 543 over the binary field Z2 was completed on a 4-processor Linux PC, using roughly 100 CPU-hours. The basic approach is a combination of the factorization software BIPOLAR and a parallel version of Cantors multiplication algorithm. The PUB-library (Paderborn University BSP library) is used for the implementation of the parallel communication.

High degree solutions of low degree equations (extended abstract)

We exhibit a class of proper hypergeometric expressions which lead to a key equation with coe cients of degree at most two and a unique solution of arbitrarily high degree in Gospers algorithm from 1978 for inde nite hypergeometric summation. We investigate similar classes for the related problems of inde nite integration and qhypergeometric summation.