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Numerical semigroups and applications. (English) Zbl 1368.20001
RSME Springer Series 1. Cham: Springer (ISBN 978-3-319-41329-7/hbk; 978-3-319-41330-3/ebook). xiv, 106 p. (2016).
The goal of this book is to introduce the theory of numerical semigroups with a view towards connections to other areas. I think of this text as being divided into three parts. In the first two chapters, the authors introduce basic properties of numerical semigroups, focusing on irreducible numerical semigroups and the more restricted class of free numerical semigroups. The third chapter, the longest and most technically demanding part of the book, shows how to associate a numerical semigroup to an irreducible polynomial in \(\mathbb{K}((x))[y]\), where \(\mathbb{K}((x))\) is the field of meromorphic series in \(x\). The authors focus on two cases of interest, the local case where \(f \in \mathbb{K}[[x]][y]\), and the case where \(f \in \mathbb{K}[x^{-1}][y]\) with the condition that \(f(x^{-1},y)\) has one place at infinity. Semigroups constructed in this way are free, and the ideas of the earlier chapters are applied here. The authors develop the theory of Newton-Puiseux exponents, characteristic sequences, intersection multiplicity, and related concepts, and show how semigroup-theoretic concepts can be related to concepts from the study of plane curves. The final part of the book consists of a chapter on minimal presentations of numerical semigroups, a way of describing a semigroup as \(\mathbb{N}^n\) modulo some relations among its generators, and a chapter on questions of non-unique factorizations in numerical semigroups.
A numerical semigroup is a submonoid of \(\mathbb{N} = \{0,1,2,\ldots\}\) with finite complement. Let \(t\) be an indeterminant and \(\mathbb{K}\) be a field. The semigroup ring over \(\mathbb{K}\) of a numerical semigroup \(S\) is \(\mathbb{K}[t^s\;|\;s \in S]\). Throughout the text, the authors explain how definitions in the theory of numerical semigroups, for example the conductor, multiplicity, genus, and type, are related to corresponding concepts from commutative algebra and algebraic geometry. The text also includes examples of computations done in the computer algebra system GAP with the numericalsgps package.
Some of the material of this book is covered in a similar way in other sources. A standard reference for the subject is the book by the second author and J. C. Rosales [Numerical semigroups. Dordrecht: Springer (2009; Zbl 1220.20047)]. Much of the material in Chapters 1, 2, and 4 can be found here. The second author has also written a survey article on computational aspects of factorization invariants that nicely complements the material of Chapter 5 [in: Multiplicative ideal theory and factorization theory. Commutative and non-commutative perspectives. Selected papers based on the presentations at the meeting ‘Arithmetic and ideal theory of rings and semigroups’, Graz, Austria, September 22–26, 2014. Cham: Springer. 159–181 (2016; Zbl 1394.20034)]. The final few pages of Chapter 5 discuss the Feng-Rao distance of an element of a numerical semigroup, one of the motivating connections between numerical semigroups and the theory of error-correcting codes. Chapter 3 develops the connection between numerical semigroups and plane curves and requires only a minimum of background in algebra and algebraic geometry. I do not know of another reference that takes a similar approach to the material of this chapter.

MSC:
20-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory
20M14 Commutative semigroups
20M05 Free semigroups, generators and relations, word problems
13F25 Formal power series rings
14H50 Plane and space curves
14G50 Applications to coding theory and cryptography of arithmetic geometry
20-04 Software, source code, etc. for problems pertaining to group theory
14-04 Software, source code, etc. for problems pertaining to algebraic geometry
Software:
Normaliz; SINGULAR
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