## Abstract

There are two aspects to the study of Gröbner bases: theory and computation. For problems which are difficult to solve by theoretical approaches, it may be possible to obtain solutions by computation, using either brute force or more elegant methods. On the other hand, for problems for which the computational methods are difficult, it may be possible to obtain solutions by a combination of theoretical insight and calculations. This is one of the attractions of Gröbner bases. Chapters 4–6 emphasized the theoretical aspect. In this chapter, we present problems and answers which utilize various software systems. It is our hope that readers will perform the calculations on these software systems while studying this chapter. Following these problems and their answers, we provide easy exercises which will help the reader to understand how to use these software systems to study or apply Gröbner bases. We will use computer algebra systems, statistical software systems, and some expert systems for polytopes and toric ideals; this covers several areas related to the theory and applications of Gröbner bases.

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## Notes

- 1.
Error messages may be displayed if Asir Contrib is not loaded since the package yang.rr uses some functions defined in Asir Contrib. If this happens, use the command import(‘‘names.rr’’);.

- 2.
The command initialIdealW accomplishes this procedure in Singular. However, there may be a bug in version 3-1-2.

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Nakayama, H., Nishiyama, K. (2013). Examples and Exercises. In: Hibi, T. (eds) Gröbner Bases. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54574-3_7

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