Abstract
In Chap. 1, we presented the theoretical foundation of Gröbner bases, and many of our computations were carried out by hand. When we want to apply Gröbner bases to practical problems, however, in most cases, we will need the help of computers. There are many mathematical software systems which support the computation of Gröbner bases, but we will often encounter cases which require careful settings or preprocessing in order to be efficient. In this chapter, we explain various methods to efficiently use a computer to compute Gröbner bases. We also present some algorithms for performing operations on the ideals realized by Gröbner bases. These operations are implemented in several mathematical software systems: Singular, Macaulay2, CoCoA, and Risa/Asir. We will illustrate the usage of these systems mainly by example.
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Notes
- 1.
The name “sugar” comes from [5]. The sugar strategy was first implemented in CoCoA.
- 2.
The versions discussed here of Macaulay2, Singular, and CoCoA are 1.4, 3.1.2, and 4.7.5, respectively. They were all executed on a KNOPPIX/Math virtual machine, and the computing time is not accurate.
- 3.
Only command line editing is available.
- 4.
If you are using CoCoA-5, you do not have to do this.
- 5.
This takes a very long time.
- 6.
If it is known that no coefficient swells will occur, then the homogenization is unnecessary.
References
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Noro, M. (2013). Computation of Gröbner Bases. In: Hibi, T. (eds) Gröbner Bases. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54574-3_3
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DOI: https://doi.org/10.1007/978-4-431-54574-3_3
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