By Martin Kreuzer
ISBN-10: 3540368043
ISBN-13: 9783540368045
Bridges the present hole within the literature among conception and genuine computation of Groebner bases and their functions. A finished advisor to either the idea and perform of computational commutative algebra, excellent to be used as a textbook for graduate or undergraduate scholars. includes tutorials on many matters that complement the fabric.
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Additional resources for Computational Commutative Algebra 1
Example text
Fm ) generates the ideal (f1 , . . , fm ). In particular, we have gcd(f1 , . . , fm ) = 1 if and only if there are elements g1 , . . , gm ∈ R such that g1 f1 +· · ·+gm fm = 1 . Proof. Since least common multiples were defined recursively, it suffices to αs 1 prove claim a) for m = 2 . Let f1 = c1 pα and f2 = c2 pβ1 1 · · · pβs s be 1 · · · ps factorizations of f1 and f2 , where c1 , c2 ∈ R are units, where αi , βj ≥ 0 , and where p1 , . . , ps ∈ R are irreducible elements representing s different max{α1 ,β1 } max{αs ,βs } equivalence classes.
FFNeg(. ), FFMult(. ), and FFInv(. ) which compute the lists representing the sums, negatives, products, and inverses of elements of K , respectively. ) f) Compute a representation of the field F16 and its multiplication table. 2 Unique Factorization Everything should be made as simple as possible, but not simpler. (Albert Einstein) In this section we discuss a fundamental property of polynomial rings over fields, namely the unique factorization property. One learns in school that for αs 1 α2 every integer n we may write n = pα 1 p2 · · · ps , where p1 , .
Xm ]r is a monomial K[x1 , . . , xm ]-submodule of K[x1 , . . , xm ]r by exhibiting an explicit monomial system of generators. Write a CoCoA function MonElim(. ) which computes this elimination module. √ i) Show that I = {f ∈ P | f i ∈ I for some i ∈ N} is the monomial ideal generated by the squarefree parts of the generators of I . Write a CoCoA function MonRadical(. ) which computes this radical ideal. Hint: The hint given here anticipates some themes explained later, starting with the next section.
Computational Commutative Algebra 1 by Martin Kreuzer
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