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Monomial ideals and their decompositions
Moore W., Rogers M., Sather-Wagstaff S., Springer International Publishing, New York, NY, 2018. 412 pp. Type: Book (978-3-319968-74-2)
Date Reviewed: May 13 2019

We usually solve problems by breaking them into smaller problems. In the predominant situations, we have linear systems that can be decomposed into several problems because these systems have a basis, that is, the system is the “product” of a number of subsystems. Although in general nonlinear systems do not have such a decomposition, the challenge for mathematicians is to extend the idea of basis to at least some of the nonlinear systems that arise in practice. This book addresses the question of how to decompose polynomial rings. A fairly obvious idea is to consider ideals. Now the hard work begins: how does one define the appropriate ideals, and how does one define the appropriate decomposition?

A ring is a structure in which addition, subtraction, and multiplication make sense. For example, the integers and integers mod 5 are both rings. An ideal is a subring that is closed under multiplication by arbitrary ring elements. If R is a ring, then the set of polynomials R [X,Y], with the variables X and Y and coefficients from R, is also a ring. All of the polynomials that are multiples of X2Y are examples of a monomial ideal of R [X,Y]. It is a monomial ideal because every element is a multiple of the monomial X2Y.

This book is divided into four parts. Part 1 gives basic results on monomial ideals. Part 2 connects monomial ideals with other areas of mathematics and applications. Part 3 looks at various ways of decomposing these ideals and how to compute these decompositions. Part 4 gives necessary background in abstract algebra and introduces the programming language Macaulay2.

Each definition includes examples of reasonably common structures that do and do not satisfy the hypotheses. This style makes the text accessible to advanced undergraduates. Contrast this with a more professional style in which definitions are given without motivation and theorems are stated in their most general form, rather than in the most applicable form. Most of the sections also include Macaulay2 programs, which show how to compute the objects or decompositions described.

I was intrigued by the idea that the vertex cover problem could be cast in terms of rings and ideals. The authors show how to use these structures to find a minimal vertex cover. Unfortunately, by minimal cover they mean a cover that becomes a non-cover when any vertex is removed. This is not the standard vertex cover problem in which one seeks the cover with the fewest vertices.

This book is not essential reading for most computer scientists, but it will be useful to those who work in symbolic computation and theory.

Reviewer:  Paul Cull Review #: CR146569 (1907-0263)
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Graph Theory (G.2.2 )
 
 
Algebraic Algorithms (I.1.2 ... )
 
 
Special-Purpose Algebraic Systems (I.1.3 ... )
 
 
Combinatorics (G.2.1 )
 
 
Nonnumerical Algorithms And Problems (F.2.2 )
 
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