By Martin Kreuzer

ISBN-10: 3540368043

ISBN-13: 9783540368045

Bridges the present hole within the literature among thought and actual computation of Groebner bases and their purposes. A finished consultant to either the idea and perform of computational commutative algebra, perfect to be used as a textbook for graduate or undergraduate scholars. comprises tutorials on many matters that complement the fabric.

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**Additional info for Computational Commutative Algebra 1**

**Sample text**

From a) we conclude that t is a multiple of an element t ∈ G2 . Using a) again, we see that t , and therefore t, is a multiple of one of the elements of G1 . e. t and t are multiples of each other. Thus t = t ∈ G2 , a contradiction. 46 1. Foundations Exercise 1. Let Γ be a commutative group. Show that in it there is only one monoideal, namely Γ itself. Exercise 2. Equip the set Γ = {0, ∞} with the “natural” addition and show that (Γ, +) is a commutative monoid in which the cancellation law does not hold.

If the polynomial is f = a0 + a1 x + · · · + ad xd , the program should return the list [a0 , . . , ad ] representing this polynomial. Hint: There is a pedestrian solution involving a For-loop and the CoCoA function CoeffOfTerm(. ) and an elegant one using the command Coefficients(. ). b) Implement also a CoCoA function ListToPoly(. ) which takes a list of numbers and constructs the corresponding univariate polynomial in the indeterminate x. Use this function and ReprPoly(. ) to convert the polynomials f1 = x4 + 3x2 − x + 1 and f2 = y 2 + 2y + 3 to lists and back.

Fm ∈ R\{0} change only by a unit if we choose a different set of representatives P for the equivalence classes of irreducible elements. We shall therefore speak of the greatest common divisor and the least common multiple of f1 , . . , fm ∈ R\{0} , as well as the squarefree part of f ∈ R\{0} , while always keeping in mind that they are unique only up to a unit. In the following, we describe some connections between greatest common divisors, least common multiples, and ideal theory. First we characterize greatest common divisors and least common multiples by divisibility properties.

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Categories: Algebraic Geometry