The solution of computational tasks from the "real world" requires high performance computations. Not limited to mathematical computing, asymptotically fast algorithms have become one of the major contributing factors in this area. Based on [4], the tutorial will give an introduction to the beauty and elegance of modern computer algebra.

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.

We analyze six algorithms for computing integral Taylor shifts for polynomials with integral coefficients.We present and analyze a new algorithm for solving the "key equation" which occurs in many rational and hypergeometric summation algorithms.In a special case, our algorithm is asymp totically faster than previously known methods.We give experimental results for our algorithms.

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.