Edge Ideals Using Macaulay2
Computer algebra systems, like Macaulay 2 [80], Singular [47], and CoCoA [39], have become essential tools for many mathematicians in commutative algebra and algebraic geometry. These systems provide a “laboratory” in which we can experiment and play with
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1 The Macaulay 2 Package EdgeIdeals Computer algebra systems, like Macaulay 2 [80], Singular [47], and CoCoA [39], have become essential tools for many mathematicians in commutative algebra and algebraic geometry. These systems provide a “laboratory” in which we can experiment and play with new ideas. From these experiments, a researcher can formulate new conjectures, and hopefully, new theorems. Computer algebra systems are especially good at dealing with monomial ideals. As a consequence, the study of edge and cover ideals is well suited to experiments using computer algebra systems. The purpose of this section is to familiarize the user with the package EdgeIdeals that was written by C. Francisco, A. Hoefel, and myself [69]. This package, written for Macaulay 2, provides a suite of functions to experiment with edge and cover ideals. Many of the results discussed in the notes have been implemented into this package. Hopefully, the tools introduced in this tutorial will be the basis of your own research results! As a final note, although I primarily discuss the EdgeIdeal package, I would recommend that your also become familiar with the packages SimplicialComplexes, written by S. Popescu, G.G. Smith, and M. Stillman (see [108]), and SimplicialDecomposability by D.W. Cook II (see [42]). The first package contains a number of useful functions related to simplicial complexes. In fact, the EdgeIdeals package requires a number of functions from this package. The SimplicialDecomposability package of D.W. Cook II is useful if you wish to study the properties of the simplicial complex associated to the edge or cover ideal of a graph.
A.V. Tuyl () Department of Mathematical Sciences, Lakehead University, Thunder Bay, Canada, ON P7B 5E1 e-mail: [email protected] A.M. Bigatti et al. (eds.), Monomial Ideals, Computations and Applications, Lecture Notes in Mathematics 2083, DOI 10.1007/978-3-642-38742-5 4, © Springer-Verlag Berlin Heidelberg 2013
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1.1 Getting Started Obviously, the first thing you need to do is install the latest version1 of Macaulay 2 on your computer. The download page is here: http://www.math.uiuc.edu/Macaulay2/Downloads/ Pick the appropriate operating system, and then follow the instructions. This may take some time and patience. I am going to assume that you have installed Macaulay 2 and now have it working. To familiarize yourself with the basic syntax and some simple examples, a good place to start is this web page: http://www.math.uiuc.edu/Macaulay2/GettingStarted/ If you have never used Macaulay 2, take a couple of minutes to try a couple of the sample sessions.
1.2 The EdgeIdeals Package Now that you have Macaulay 2 installed, we want to load the EdgeIdeals package. If you are using a current version of Macaulay 2 (i.e., a version 1:2), then this package should already be included with your installation of Macaulay 2, and it simply has to be installed. Remark 1. If you have an older version, or if your version does not include this package, you should first download the s
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