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By W. J. Thron

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Extra resources for Analytic Theory of Continued Fractions II

Sample text

En d´eduire un algorithme pour d´eterminer le satur´e de I par J. 18. Quotient des id´ eaux de K[x]. Soient I et J = (h1 , . . , ht ) deux id´eaux de K[x]. 44 M. Elkadi & B. Mourrain R´esolution des syst`emes polynomiaux 1. Si I ∩ (hi ) = (g1 , . . , gr ), montrer que I : (hi ) = 2. Montrer que I : J = t i=1 g1 gr . , hi hi I : (hi ) . 3. En d´eduire un algorithme pour calculer I : J. 19. Bases de Gr¨ obner des sous-modules de K[x]m . ´ l’algorithme de division de f par la famille 1. Soient f, f1 , .

Fs ) de K[x]. 1. Appartenance d’un polynˆ ome ` a un id´ eal. — Comment peut-on tester l’appartenance d’un polynˆ ome f `a I ? 9, f ∈ I si, et seulement si, NG (f ) = 0. Donc f ∈ I ⇐⇒ f se r´eduit a` z´ero par G. 30. Une question int´eressante, sous-jacente au probl`eme de l’appartenance d’un polynˆ ome f `a l’id´eal I, est celle de la repr´esentation : si f ∈ I, d´eterminer des polynˆomes q1 , . . , qs tels que f = q1 f1 + · · · + qs fs . Pour cela, on peut diviser f par G = {g1 , . . , gt }, puis exprimer chaque gi en fonction de f1 , .

Si {f1 , . . , fs } est une base de Gr¨obner, montrer que Syz(f1 , . . , fs ) est engendr´e par σij = ppcm m(fi ), m(fj ) ej ei − t(fi ) t(fj ) s − k=1 qij,k ek , 1 ≤ i < j ≤ s, o` u les qij,k sont les quotients de la division de S(fi , fj ) par {f1 , . . , fs }. 2. Soit G = {g1 , . . , gt } une base de Gr¨ obner de (f1 , . . , fs ). Notons par f et g les vecteurs de composantes f1 , . . , fs et g1 , . . , gt , M la matrice obtenue par la division de chaque fi par G et qui satisfait f = gM , N la matrice obtenue lors de la construction de G par l’algorithme de Buchberger et qui v´erifie g = f N .

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