The membership problem for polynomial ideals in terms of residue currents

Andersson, Mats

Annales de l'Institut Fourier, Tome 56 (2006) no. 1, pp. 101-119.

On obtient un lien entre l’annulation d’un courant résidu défini globalement sur ${\mathbb{P}}^n$ et la solvabilité du problème d’appartenance pour les idéaux de polynômes où l’on contrôle les degrés des polynômes. Plusieurs théorèmes classiques se déduisent comme cas particuliers, comme par exemple le théorème de Max Noether, dont on obtient de plus une généralisation. On trouve également des liens avec des versions effectives du Nullstellensatz. On donne aussi des représentations intégrales explicites des solutions.

We find a relation between the vanishing of a globally defined residue current on ${\mathbb{P}}^n$ and solution of the membership problem with control of the polynomial degrees. Several classical results appear as special cases, such as Max Nöther’s theorem, for which we also obtain a generalization. Furthermore there are some connections to effective versions of the Nullstellensatz. We also provide explicit integral representations of the solutions.

Classification : 32B99, 32A27, 14E99
Keywords: membership problem, polynomial ideal, residue current, integral representation
Mot clés : problème d’appartenance, idéaux de polynômes, courant résidu, représentation intégrale

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