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By Eduard L. Stiefel

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Example text

Une Γ-base de l’id´eal I est un syst`eme de g´en´erateurs de I. D´emonstration. La r´eduction a` 0 de tout ´el´ement p ∈ I par une Γ-base G = {g1 , . . , gt } implique une d´ecomposition de la forme p = ti=1 hi gi , avec hi ∈ K[x] . Le syst`eme G engendre bien l’id´eal I. 11. Le reste r de la division de f ∈ K[x] par une Γ-base G de I est unique. Il est appel´e la forme normale de f par rapport a ` G, et not´e NG (f ). D´emonstration. Soient r1 et r2 deux restes de la division de f par G. Comme r1 − r2 est r´eduit par rapport a` G et r1 − r2 ∈ I, r1 − r2 = 0.

Hl ) deux id´eaux de K[x]. Notons 1 = (1, 1) , F1 = (f1 , 0) , . . , Fs = (fs , 0) , H1 = (0, h1 ) , . . , Hl = (0, hl ), π1 : (g1 , . . , gs+l+1 ) ∈ K[x]s+l+1 → g1 ∈ K[x]. 1. Montrer que I ∩ J = π1 Syz(1, F1 , . . , Fs , H1 , . . , Hl ) . 2. Montrer que si le K[x]-module Syz(1, F1 , . . , Fs , H1 , . . , Hl ) est engendr´e par G1 , . . , Gr , alors l’id´eal I ∩ J = π1 (G1 ), . . , π1 (Gr ) . 3. En d´eduire un algorithme pour calculer l’intersection des id´eaux de K[x]. 23. Soient f0 f1 = = 2 x − 4 xy + 4 xy 2 − 2 x2 + 4 x2 y − 4 x2 y 2 + 2 y − 2 y 2 4 xy − 4 xy 2 f2 f3 = = 2 y − 2 y 2 − 8 xy + 10 xy 2 + 8 x2 y − 10 x2 y 2 2 xy 2 − 2 x2 y 2 .

2. Montrer que la vari´et´e Z(J ∩ K[x]) est irr´eductible. 3. En d´eduire un algorithme pour passer d’une repr´esentation param´etr´ee ⎧ f1 (t1 , . . , ts ) ⎪ ⎪ ⎪ x1 = ⎪ ⎪ d 1 (t1 , . . , ts ) ⎨ .. ⎪ ⎪ ⎪ fn (t1 , . . , ts ) ⎪ ⎪ ⎩ xn = dn (t1 , . . , ts ) omes) d’une vari´et´e alg´ebrique V de Kn ` a (les fi , di , i = 1, . . e. V = Z(g1 , . . , gt ), avec g1 , . . , gt ∈ K[x]). 17. Satur´ e d’un id´ eal par un autre id´ eal. Soient I et J = (g1 , . . , gt ) deux id´eaux de K[x]. L’id´eal satur´e de I par J est (I : J ∗ ) = ∪i∈N (I : J i ) = {f ∈ K[x] : il existe m ∈ N , f J m ⊂ I}.

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