University of Paderborn Department of Mathematics Diploma Thesis ...

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6described in Chapter 4.Additionally, with the friendly permission of the MuPAD-Research-Group and the companySciface Software GmbH, several versions of MuPAD could be added to the CD. The usermay find a MuPAD-Light version for the use of MuPAD under Microsoft Windows and aMuPAD-Light version for the use under Unix respectively Linux. They can be used forfree by anyone, who is working in an academic area, and contain the whole mathematicalfunctionality of MuPAD (the only disadvantages towards the official versions of MuPADare the lack of graphics and the lack of a user interface). For details, we refer to the websiteshttp://www.mupad.de/schule+studium/downloadandhttp://www.sciface.com.There one finds additional information on technical support and the latest versions of theMuPAD documentation files in HTML-format and in PDF-format.
Chapter 1Notations and PrerequisitesWe want to recall some basic definitions and, according to this thesis, some central results.The more standard definitions and results will be mentioned within the text and will not benumerated explicitly (there is a quick reference of notations in the Glossary of Notations,see page 127). Definitions and results, which may not be known in general and which serveto develop the main results of this special thesis, will be stated in the usual, formal wayand be numerated to make it easy to refer to them.1.1 General prerequisites and terminologyLet K be a field of characteristic zero, and let R := K[x 0 , . . . , x n ] denote the polynomialring in n + 1-Variables x 0 , . . . , x n . Each element f ∈ R can be uniquely written in the formk∑f = r i · m i , where r i ∈ K and m i = x a i00 · x a i11 · . . . · x a inn ∈ R with a i0 , a i1 , . . . , a in ∈ N 0 .i=0For each 0 ≤ i ≤ k we call r i · m i a term of the polynomial f, r i the coefficient of the termn∑and m i a monomial. The degree of a monomial m = x a 00 · . . . · x ann ∈ R is deg m := a i .The degree of a polynomial f =k∑r i ·m i ∈ R is defined to be the maximum of the degreesi=0of its monomial components m i , that is deg f := max{deg m i | i = 0, . . . , k}. We call f ahomogeneous polynomial or homogeneous of degree d, if there is d ∈ N 0 such that all termsof f have degree d.Thus, the polynomial ring R becomes a graded ring over K, graded by the degree. In eachdegree d, the set of all polynomials of degree d forms a vector space over K.In some situations, in particular in connection with exact sequences, we will have to shiftthe grading of R. For k ∈ N we write R(−k) to denote that every element f ∈ R, deg f = d,has degree d − k when viewed as an element of R(−k).We will denote the set of all monomials of R by M, i.e.M := {m | m ∈ R, m is a monomial} .7i=0
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[13] F.S. Macaulay, Some properties
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