CoCoA

1 What is CoCoA?

CoCoA is a special-purpose system for doing Computations in Commutative Algebra. It runs on the following platforms: SUN running Solaris, HP, SGI running IRIX, DEC Alpha, PC running or DOS or Windows, PPC running MacOS or LinuxPPC. CoCoA is one of the products of a research team in Computer Algebra, whose members are: Lorenzo Robbiano, Gianfranco Niesi, John Abbott, Anna Bigatti, Massimo Caboara, Martin Kreuzer, David Perkinson, Alessandro Polverini, Antonio Capani, Volker Augustin, Arndt Wills, and occasionally other researchers and students. The system includes complete on-line help (also available as an html-manual). CoCoA is freely available software for research and educational purposes: the latest version is CoCoA 4.1 (May 2001) which can be obtained by anonymous ftp from http://cocoa.dima.unige.it as well as from the mirror sites http://www.physik.uni-regensburg.de/~krm03530/ http://www.reed.edu/mirrors/cocoa

2 The main features of CoCoA

CoCoA’s principal area of expertise is that of operations over commutative rings of polynomials. For exam- ple, it can readily compute Gr¨obnerbases, syzygies and minimal free resolutions ([CDNR]), intersections, divisions ([CT]), the radical of an ideal ([CCT]), the ideals of 0-dimensional schemes ([ABKR], [AKR]), Poincar´eseries and Hilbert functions ([B, CDR]), factorization of polynomials ([A]), toric ideals ([BLR]). The capabilities of CoCoA and the flexibility of its use are further enhanced by the dedicated high-level programming language CoCoAL. For convenience, the system offers a textual interface, an mode, and a graphical user interface common to most platforms.

3 The users of CoCoA

Currently CoCoA is used by researchers in several countries. Most of them are Commutative Algebraists and Algebraic Geometers, but also people working in different areas such as Analysis and Statistics (see [R]) have already benefitted from our system. CoCoA is also used as the main system for teaching advanced courses in several Universities. Besides Italy, the most intensive use is by Tomas Recio at the University of Santander (Spain), by Anthony Geramita at Queen’s University (Canada), by Martin Kreuzer at the University of Regensburg (Germany), by Dave Perkinson at Reed College (USA), and by Marie Vitulli at the University of Oregon (USA). CoCoA was one of the few systems to have been invited to participate in the Special Session on Mathematical Software at ICM’98 and at ECM’00. It is also mentioned in some of the most widely used text books in Computational Algebra (see for instance [AL] pp. 275–276, and [CLO] pp. 493–494), and plays a major role in the book [KR].

4 The future of CoCoA

Aside from the normal continual development, a number of more specific plans are afoot to improve and extend CoCoA: the choice of coefficients is to be widened to handle parameters and finite algebraic extensions; and the mathematical core will be made available as a software library facilitating integration into other systems. References

[A] J. Abbott, Univariate factorization over the integers, Preprint (1998). [ABKR] J. Abbott, A.M. Bigatti, M. Kreuzer, L. Robbiano Computing Ideals of Points, J. Symbolic Comput., To appear. [AKR] J. Abbott, M. Kreuzer, L. Robbiano Computing Zero-Dimensional Schemes, Preprint (2000). [AL] W. W. Adams, P. Loustaunau, An Introduction to Gr¨obnerBases, Graduate Studies in Math- ematics: Amer. Math. Soc., Providence, R.I. (1994). [B] A.M. Bigatti, Computation of Hilbert-Poincar´eSeries, J. Pure Appl. Algebra, 119/3, 237–253 (1997). [BLR] A.M. Bigatti, R. La Scala, L. Robbiano, Computing Toric Ideals, J. Symbolic Comput., 27, 351–365, (1999). [CCT] M. Caboara, P. Conti, and C. Traverso, Yet Another Ideal Decomposition Algorithm, AAECC- 12, Springer LNCS 1255, 39–54, (1997). [CD] A. Capani, G. De Dominicis, Web Algebra, In Proc. of WebNet 96. Association for the Ad- vancement of Computing in Education (AACE) Charlottesville, USA, (1996). [CDNR] A. Capani, G. De Dominicis, G. Niesi, L. Robbiano, Computing Minimal Finite Free Resolu- tions, J. Pure Appl. Algebra, 117/118, 105–117 (1997). [CDR] M. Caboara, G. De Dominicis, L. Robbiano, Multigraded Hilbert Functions and Buchberger Algorithm, In Proc. ISSAC ’96, 72–78 (1996), Y.N. Lakshman, editor, New York. ACM Press. [CLO] D. Cox, J. Little, D. O’Shea, Ideals, Varieties, and Algorithms , Springer-Verlag, New York (1992) [CN] A. Capani, G. Niesi, The CoCoA 3 Framework for a Family of Buchberger-like Algorithms, In Gr¨obnerBases and Applications (Proc. of the Conf. 33 Years of Gr¨obnerBases), London Math. Soc. Lecture Notes Series, Vol 251, B. Buchberger and F. Winkler eds, Cambridge University Press, 338–350 (1998). [CNR] A. Capani, G. Niesi, L. Robbiano, Some Features of CoCoA 3, Comput. Sci. J. of Moldova 4, 296–314 (1996). [CT] M. Caboara, C. Traverso, Efficient Algorithms for Ideal Operations, In Proc. ISSAC ’98, 147– 152 (1998), ACM Press. [GMNRT] A. Giovini, T. Mora, G. Niesi, L. Robbiano, C. Traverso, “One sugar cube, please” or selection strategies in the Buchberger algorithm, In Proc. ISSAC ’91, Stephen M. Watt, editor, New York, ACM Press, 49–54 (1991). [GN] A. Giovini and G. Niesi, CoCoA: a user-friendly system for commutative algebra, In Design and Implementation of Symbolic Computation Systems – International Symposium DISCO’90, Lecture Notes in Comput. Sci., 429, 20–29, Berlin, Springer Verlag (1990). [KR] M. Kreuzer, L. Robbiano, Computational Commutative Algebra 1, Springer (2000). [R] L. Robbiano, Gr¨obnerBases and Statistics, In Gr¨obnerBases and Applications (Proc. of the Conf. 33 Years of Gr¨obnerBases), London Math. Soc. Lecture Notes Series, Vol 251, B. Buchberger and F. Winkler eds, Cambridge University Press, 179–204 (1998).

May 2001