Backelin, Jörgen; Cojocaru, Svetlana; Ufnarovski, Victor The computer algebra package Bergman: Current state. (English) Zbl 1084.13503 Herzog, Jürgen (ed.) et al., Commutative algebra, singularities and computer algebra. Proceedings of the NATO advanced research workshop, Sinaia, Romania, September 17–22, 2002. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-1486-4/hbk; 1-4020-1487-2/pbk). NATO Sci. Ser. II, Math. Phys. Chem. 115, 75-100 (2003). In commutative algebra and algebraic geometry symbolic computation software has played a very important role and has accelerated algorithmic developments. The paper under review is an introduction to “Bergman”, a symbolic computation package that allows the computation of (commutative and) non-commutative Gröbner bases and Hilbert series in graded algebras and modules over them. In addition to non-commutative Gröbner bases, the user can compute Anick resolutions, Betti numbers, and Poincaré series, but only in the non-commutative setting. Bergman is not a stand-alone package, but can be run under Reduce. It is public domain software written in Standard Lisp and runs on Sun-Sparc stations, Alpha stations under DEC ODF and Linux, as well as on PCs under Windows NT 95 or later.For the entire collection see [Zbl 1024.00025]. Reviewer: Reinhard C. Laubenbacher (Blacksburg) Cited in 3 Documents MSC: 13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 18G10 Resolutions; derived functors (category-theoretic aspects) 16S15 Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting) 16Z05 Computational aspects of associative rings (general theory) 68W30 Symbolic computation and algebraic computation Keywords:Gröbner basis; Betti numbers; Hilbert series; Poincaré series; Hochschild homology; Anick resolution Software:BERGMAN PDFBibTeX XMLCite \textit{J. Backelin} et al., NATO Sci. Ser. II, Math. Phys. Chem. 115, 75--100 (2003; Zbl 1084.13503)