The membership problem for polynomial ideals in terms of residue currents  [ Le problème d’appartenance pour les idéaux de polynômes en termes de courants résidus ]
Annales de l'Institut Fourier, Tome 56 (2006) no. 1, p. 101-119
On obtient un lien entre l’annulation d’un courant résidu défini globalement sur 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 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.
DOI : https://doi.org/10.5802/aif.2174
Classification:  32B99,  32A27,  14E99
Mots clés: problème d’appartenance, idéaux de polynômes, courant résidu, représentation intégrale
@article{AIF_2006__56_1_101_0,
     author = {Andersson, Mats},
     title = {The membership problem for polynomial ideals in terms of residue currents},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {56},
     number = {1},
     year = {2006},
     pages = {101-119},
     doi = {10.5802/aif.2174},
     mrnumber = {2228682},
     zbl = {1092.32002},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2006__56_1_101_0}
}
Andersson, Mats. The membership problem for polynomial ideals in terms of residue currents. Annales de l'Institut Fourier, Tome 56 (2006) no. 1, pp. 101-119. doi : 10.5802/aif.2174. http://www.numdam.org/item/AIF_2006__56_1_101_0/

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