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Über dieses Buch

Featuring up-to-date coverage of three topics lying at the intersection of combinatorics and commutative algebra, namely Koszul algebras, primary decompositions and subdivision operations in simplicial complexes, this book has its focus on computations. "Computations and Combinatorics in Commutative Algebra" has been written by experts in both theoretical and computational aspects of these three subjects and is aimed at a broad audience, from experienced researchers who want to have an easy but deep review of the topics covered to postgraduate students who need a quick introduction to the techniques. The computational treatment of the material, including plenty of examples and code, will be useful for a wide range of professionals interested in the connections between commutative algebra and combinatorics.



Koszul Algebras and Computations

A Koszul algebra R is a \(\mathbb{N}\)-graded K-algebra whose residue field K has a linear free resolution as an R-module. Many papers and lectures have been given on this topic, so here we collect various properties and facts which are related to being a Koszul algebra, and illustrate their mutual implications or counter-examples.
In addition we explain how one can investigate computationally these many aspects, some of which would seem to be intrinsically intractable, and we show many examples by using CoCoA-5.
Anna M. Bigatti, Emanuela De Negri

Primary Decompositions

with Sections on Macaulay2 and Networks
This chapter contains three major sections, each one roughly corresponding to a lecture. The first section is on computing primary decompositions, the second one is more specifically on binomial ideals, and the last one is on some primary decomposition questions in algebraic statistics and networks.
Irena Swanson, Eduardo Sáenz-de-Cabezón

Combinatorics and Algebra of Geometric Subdivision Operations

In the subsequent sections we survey results from combinatorics, discrete geometry and commutative algebra concerning invariants and properties of subdivisions of simplicial complexes. For most of the time we are interested in deriving results that hold for specific subdivision operations that are motivated from combinatorics, geometry and algebra. In particular, we study barycentric, edgewise and interval subdivisions (see Sect. 3 for the respective definitions). Even though we mention some suspicion that part of the results we present may only be a glimpse of what is true for general subdivision operations we do not focus on this aspect. In particular, we are quite sure that some asymptotic results and some convergence results from Sect. 9 are just instances of more general phenomena. Overall, retriangulations are subtle geometric operations and we refer the reader to the book De Loera et al. (Algorithms and Computation in Mathematics. Springer, Heidelberg, 2010) for a comprehensive introduction. Since our focus lies on specific constructions we make only little use of the theory from De Loera et al. (Algorithms and Computation in Mathematics. Springer, Heidelberg, 2010). Nevertheless, we are convinced that if one wants to go beyond specific subdivision operations it will become inevitable to dig deeper into the theory of triangulations.
Fatemeh Mohammadi, Volkmar Welker


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