Decker, W.; Schreyer, F.-O. Computational algebraic geometry today. (English) Zbl 1011.14019 Ciliberto, Ciro (ed.) et al., Applications of algebraic geometry to coding theory, physics and computation. Proceedings of the NATO advanced research workshop, Eilat, Israel, February 25-March 1, 2001. Dordrecht: Kluwer Academic Publishers. NATO Sci. Ser. II, Math. Phys. Chem. 36, 65-119 (2001). From the abstract: In this article we give a survey on computational algebraic geometry today. It is not our goal to guide research in this area, but to help algebraic geometers to decide whether nowadays’ computational techniques and computer algebra systems provide useful tools for their own research. To illustrate some of the main techniques we focus on rather small examples. But we also give hints on the actual size of the computations which can be carried through today, and we quote further survey articles for more information in this direction.” Examples feature MACAULAY2, SINGULAR, the MAPLE packages SCHUBERT and CASA, and SURF, a system for visualizing curves and surfaces. Topics covered include basic applications of Gröbner bases (e.g. division, elimination, Hilbert polynomials), local rings, homological algebra and computation of coherent cohomology, \(D\)-module computations, primary decomposition, normalization, deformation theory, invariant rings, construction of special varieties. There is a large bibliography.For the entire collection see [Zbl 0971.00013]. Reviewer: T.G.Berry (Miami) MSC: 14Q05 Computational aspects of algebraic curves 14Q10 Computational aspects of algebraic surfaces 14-04 Software, source code, etc. for problems pertaining to algebraic geometry 13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 14Q15 Computational aspects of higher-dimensional varieties Keywords:Bibliography; computer algebra systems; Gröbner bases; ideal membership; Hilbert function; elimination; Milnor numbers; Beilinson monads; \(D\)-modules; primary decomposition; normalization; Puiseux expansion; rational parametrization; deformations; invariant rings; special varieties; intersection theory; syzygy conjectures; Zariski’s conjecture; visualisation; complexity Software:Macaulay2; Maple; SINGULAR; CASA; Surf; schubert PDFBibTeX XMLCite \textit{W. Decker} and \textit{F. O. Schreyer}, NATO Sci. Ser. II, Math. Phys. Chem. 36, 65--119 (2001; Zbl 1011.14019)