By Gert-Martin Greuel, Visit Amazon's Gerhard Pfister Page, search results, Learn about Author Central, Gerhard Pfister, , O. Bachmann, C. Lossen, H. Schönemann

ISBN-10: 3540735410

ISBN-13: 9783540735410

From the experiences of the 1st edition:

''It is unquestionably no exaggeration to claim that … a novel advent to Commutative Algebra goals to guide an extra level within the computational revolution in commutative algebra … . one of the nice strengths and such a lot precise positive factors … is a brand new, thoroughly unified remedy of the worldwide and native theories. … making it the most versatile and best structures of its type....another power of Greuel and Pfister's ebook is its breadth of insurance of theoretical themes within the parts of commutative algebra closest to algebraic geometry, with algorithmic remedies of virtually each topic....Greuel and Pfister have written a particular and hugely helpful booklet that are meant to be within the library of each commutative algebraist and algebraic geometer, specialist and amateur alike.''

J.B. Little, MAA, March 2004

The moment version is considerably enlarged via a bankruptcy on Groebner bases in non-commtative jewelry, a bankruptcy on attribute and triangular units with functions to basic decomposition and polynomial fixing and an appendix on polynomial factorization together with factorization over algebraic box extensions and absolute factorization, within the uni- and multivariate case.

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Extra resources for A Singular Introduction to Commutative Algebra

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1) A finite set G ⊂ R is called a standard basis of I if G ⊂ I, and L(I) = L(G) . That is, G is a standard basis, if the leading monomials of the elements of G generate the leading ideal of I, or, in other words, if for any f ∈ I {0} there exists a g ∈ G satisfying LM(g) | LM(f ). (2) If > is global, a standard basis is also called a Gr¨ obner basis. (3) If we just say that G is a standard basis, we mean that G is a standard basis of the ideal G R generated by G. Let > be any monomial ordering on Mon(x1 , .

5. Let > be any monomial ordering: (1) For f ∈ K[x]> choose u ∈ K[x] such that LT(u) = 1 and uf ∈ K[x]. We define LM(f ) := LM(uf ), LC(f ) := LC(uf ), LT(f ) := LT(uf ), LE(f ) := LE(uf ), and tail(f ) = f − LT(f ). (2) For any subset G ⊂ K[x]> define the ideal L> (G) := L(G) := LM(g) | g ∈ G {0} K[x] . L(G) ⊂ K[x] is called the leading ideal of G. 6. 5 (1) are independent of the choice of u. (2) Since K[x]> ⊂ K[x] x ⊂ K[[x]], where K[[x]] denotes the formal power series ring (cf. 1), we may consider f ∈ K[x]> as a formal power series.

Fk , that is, ϕ(xi ) = fi , then ϕ is surjective if and only if y1 , . . , ym are contained in the subring Im ϕ = K[f1 , . . , fm ] of K[y1 , . . , ym ]. This fact is used in Singular to check surjectivity. We shall explain the algorithms for checking injectivity, surjectivity, bijectivity of a ring map in Chapter 2. lib. 3 (properties of ring maps). lib and which are also loaded. We suppress this message. We test injectivity using the procedure is_injective, then we compute the kernel by using the procedure alg_kernel (which displays the kernel, an object of the preimage ring, as a string).

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A Singular Introduction to Commutative Algebra by Gert-Martin Greuel, Visit Amazon's Gerhard Pfister Page, search results, Learn about Author Central, Gerhard Pfister, , O. Bachmann, C. Lossen, H. Schönemann

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