Algorithms in Symbolic Computation

The development of computer technology has brought forth a renaissance of algorithmic mathematics which gave rise to the creation of new disciplines like Computational Mathematics. Symbolic Computation, which constitutes one of its major branches, is the main research focus of the Research Institute for Symbolic Computation (RISC). In Section 1, author P. Paule, one finds an introduction to the theme together with comments on history as well as on the use of the computer for mathematical discovery and proving. The remaining sections of the chapter present more detailed descriptions of hot research topics currently pursued at RISC. In Section 2 the inventor of Gröbner Bases, B. Buchberger, describes basic notions and results, and underlines the principal relevance of Gröbner bases by surprising recent applications. Section 3, author F. Winkler, gives an introduction to algebraic curves; a summary of results in theory and applications (e.g., computer aided design) is given. Section 4, author M. Kauers, reports on computer generated progress in lattice paths theory finding applications in combinatorics and physics. Section 5, author C. Schneider, provides a description of an interdisciplinary research project with DESY (Deutsches Elektronen-Synchrotron, Berlin/Zeuthen). Section 6, author E. Kartashova, describes the development of Nonlinear Resonance Analysis, a new branch of mathematical physics.