Constructive Ideal Theory
In this chapter we will provide the basic algorithmic tools which will be used in later chapters. More precisely, we introduce some algorithms of constructive ideal theory, almost all of which are based on Gröbner bases. As the reader will find out, these algorithms and thus Gröbner bases literally permeate this book. When Sturmfels’ book  was published, not much introductory literature on Gröbner bases and their applications was available. In contrast, we now have the books by Becker and Weispfenning , Adams and Loustaunau , Cox et al. , Vasconcelos , Cox et al. , Kreuzer and Robbiano , and a chapter from Eisenbud . This list of references could be continued further. We will draw heavily on these sources and restrict ourselves to giving a rather short overview of the part of the theory that we require. The algorithms introduced in Sections 1.1–1.3 of this chapter have efficient implementations in various computer algebra systems, such as CoCoA , MACAULAY (2) , MAGMA , or SINGULAR , to name just a few, rather specialized ones. The normalization algorithm explained in Section 1.6 is implemented in MACAULAY and SINGULAR.
KeywordsNormal Form Prime Ideal Polynomial Ring Hilbert Series Monomial Ideal
Unable to display preview. Download preview PDF.