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Mats Andersson
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, 56 no. 1 (2006), p. 101-119, doi: 10.5802/aif.2174
Article PDF | Reviews MR 2228682 | Zbl 1092.32002
Class. Math.: 32B99, 32A27, 14E99
Keywords: membership problem, polynomial ideal, residue current, integral representation
[1] M. Andersson, “Integral representation with weights I”, Math. Ann. 326 (2003), p. 1-18 Article | MR 1981609 | Zbl 1024.32005
[2] M. Andersson, “Ideals of smooth functions and residue currents”, J. Funtional Anal. 212 (2004), p. 76-88 Article | MR 2065238 | Zbl 1056.32006
[3] M. Andersson, “Residue currents and ideals of holomorphic functions”, Bull. Sci. Math. 128 (2004), p. 481-512 Article | MR 2074610 | Zbl 02153822
[4] C. Berenstein, R. Gay, A. Vidras & A. Yger, Residue Currents and Bézout Identities, Birkhäuser, 1993 MR 1249478 | Zbl 0802.32001
[5] C. Berenstein & B. A. Taylor, “Interpolation problems in $\mathbb{C}^{n}$ with applications to harmonic analysis”, J. Analyse Math. 38 (1980), p. 188-254 MR 600786 | Zbl 0464.42003
[6] C. Berenstein & A. Yger, “Effective Bézout Identities in ${\mathcal{Q}}[z_1,\ldots ,z_n]$”, Acta Mathematica 166 (1991), p. 69-120 Article | MR 1088983 | Zbl 0724.32002
[7] C. Berenstein & A. Yger, “Analytic residue theory in the non-complete intersection case”, J. Reine Angew. Math. 527 (2000), p. 203-235 Article | MR 1794023 | Zbl 0960.32004
[8] B. Berndtsson, “A formula for division and interpolation”, Math. Ann. 263 (1983), p. 113-160 Article | MR 707239 | Zbl 0499.32013
[9] B. Berndtsson, “Integral formulas on projective space and the Radon transform of Gindikin-Henkin-Polyakov”, Publ. Mat. 32 (1988), p. 7-41 MR 939766 | Zbl 0659.32008
[10] B. Berndtsson & M. Andersson, “Henkin-Ramirez formulas with weights”, Ann. Inst. Fourier 32 (1982), p. 91-110 Cedram | MR 688022 | Zbl 0466.32001
[11] J. Briançon & H. Skoda, “Sur la clôture intégrale d’un idéal de germes de fonctions holomorphes en un point de $\mathbb{C}^{n}$”, C. R. Acad. Sci. Paris Sér. A 278 (1974), p. 949-951 MR 340642 | Zbl 0307.32007
[12] W. D. Brownawell, “Bounds for the degrees in Nullstellensatz”, Ann. of Math. 126 (1987), p. 577-592 Article | MR 916719 | Zbl 0641.14001
[13] W. D. Brownawell, “A pure prime product version of the Hilbert Nullstellensatz”, Mich. Math. J. 45 (1998), p. 581-597 Article | MR 1653279 | Zbl 0964.14002
[14] J-P Demailly, Complex Analytic and Differential Geometry, monograph, Grenoble, 1997
[15] A. Dickenstein & C. Sessa, “Canonical representatives in moderate cohomology”, Invent. Math. 80 (1985), p. 417-434 Article | MR 791667 | Zbl 0556.32005
[16] L. Ein & R. Lazarsfeld, “A geometric effective Nullstellensatz”, Invent. math. 135 (1999), p. 427-448 Article | MR 1705839 | Zbl 0944.14003
[17] A. Fabiano, A. Ploski & P. Tworzewski, “Effective Nullstellensatz for strictly regular sequences”, Univ. Iagel. Acta Math. 38 (2000), p. 163-167 MR 1812110 | Zbl 01797615
[18] P. Griffiths & J. Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978 MR 507725 | Zbl 0408.14001
[19] M. Hickel, “Solution d’une conjecture de C. Berenstein-A. Yger et invariants de contact à l’infini”, Ann. Inst. Fourier 51 (2001), p. 707-744 Cedram | MR 1838463 | Zbl 0991.13009
[20] Z. Jelonek, “On the effective Nullstellensatz”, preprint, 2004 Zbl 1087.14003
[21] J. Kollár, “Sharp effective Nullstellensatz”, J. American Math. Soc. 1 (1988), p. 963-975 Article | MR 944576 | Zbl 0682.14001
[22] F.S. Macaulay, The algebraic theory of modular systems, Cambridge Univ. Press, Cambridge, 1916 MR 1281612 | Zbl 0802.13001 | JFM 46.0167.01
[23] M. Nöther, “Über einen Satz aus der Theorie der algebraischen Functionen”, Math. Ann. (1873), p. 351-359 Article | MR 1509826 | JFM 05.0348.02
[24] M. Passare, “Residues, currents, and their relation to ideals of holomorphic functions”, Math. Scand. 62 (1988), p. 75-152 MR 961584 | Zbl 0633.32005
[25] M. Passare, A. Tsikh & A. Yger, “Residue currents of the Bochner-Martinelli type”, Publ. Mat. 44 (2000), p. 85-117 MR 1775747 | Zbl 0964.32003
[26] B. Shiffman, “Degree bounds for the division problem in polynomial ideals”, Michigan Math. J. 36 (1989), p. 163-171 Article | MR 1000520 | Zbl 0691.12010
[27] H. Skoda, “Application des techniques $L^{2}$ à la théorie des idéaux d’une algèbre de fonctions holomorphes avec poids”, Ann. Sci. École Norm. Sup. 5 (1972), p. 545-579 Numdam | MR 333246 | Zbl 0254.32017
[28] B. Teissier, “Résultats récents d’algèbre commutative effective”, Séminaire Bourbaki 1989/90 (1990) Numdam | Zbl 0743.13017
[29] A. Tsikh, “Multidimensional residues and their applications”, Transl. Amer. Math. Soc. 103 (1992) MR 1181199 | Zbl 0758.32001
[30] A. Tsikh & A. Yger, “Residue currents. Complex analysis”, J. Math. Sci. (N. Y.) 120 (2004), p. 1916-1971 Article | MR 2085502 | Zbl 02170265
[31] A. Vidras & A. Yger, “On some generalizations of Jacobi’s residue formula”, Ann. Sci. École Norm. Sup. 34 (2001), p. 131-157 Numdam | MR 1833092 | Zbl 0991.32003
Mats Andersson
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, 56 no. 1 (2006), p. 101-119, doi: 10.5802/aif.2174
Article PDF | Reviews MR 2228682 | Zbl 1092.32002
Class. Math.: 32B99, 32A27, 14E99
Keywords: membership problem, polynomial ideal, residue current, integral representation
Résumé - Abstract
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.
Bibliography
[2] M. Andersson, “Ideals of smooth functions and residue currents”, J. Funtional Anal. 212 (2004), p. 76-88 Article | MR 2065238 | Zbl 1056.32006
[3] M. Andersson, “Residue currents and ideals of holomorphic functions”, Bull. Sci. Math. 128 (2004), p. 481-512 Article | MR 2074610 | Zbl 02153822
[4] C. Berenstein, R. Gay, A. Vidras & A. Yger, Residue Currents and Bézout Identities, Birkhäuser, 1993 MR 1249478 | Zbl 0802.32001
[5] C. Berenstein & B. A. Taylor, “Interpolation problems in $\mathbb{C}^{n}$ with applications to harmonic analysis”, J. Analyse Math. 38 (1980), p. 188-254 MR 600786 | Zbl 0464.42003
[6] C. Berenstein & A. Yger, “Effective Bézout Identities in ${\mathcal{Q}}[z_1,\ldots ,z_n]$”, Acta Mathematica 166 (1991), p. 69-120 Article | MR 1088983 | Zbl 0724.32002
[7] C. Berenstein & A. Yger, “Analytic residue theory in the non-complete intersection case”, J. Reine Angew. Math. 527 (2000), p. 203-235 Article | MR 1794023 | Zbl 0960.32004
[8] B. Berndtsson, “A formula for division and interpolation”, Math. Ann. 263 (1983), p. 113-160 Article | MR 707239 | Zbl 0499.32013
[9] B. Berndtsson, “Integral formulas on projective space and the Radon transform of Gindikin-Henkin-Polyakov”, Publ. Mat. 32 (1988), p. 7-41 MR 939766 | Zbl 0659.32008
[10] B. Berndtsson & M. Andersson, “Henkin-Ramirez formulas with weights”, Ann. Inst. Fourier 32 (1982), p. 91-110 Cedram | MR 688022 | Zbl 0466.32001
[11] J. Briançon & H. Skoda, “Sur la clôture intégrale d’un idéal de germes de fonctions holomorphes en un point de $\mathbb{C}^{n}$”, C. R. Acad. Sci. Paris Sér. A 278 (1974), p. 949-951 MR 340642 | Zbl 0307.32007
[12] W. D. Brownawell, “Bounds for the degrees in Nullstellensatz”, Ann. of Math. 126 (1987), p. 577-592 Article | MR 916719 | Zbl 0641.14001
[13] W. D. Brownawell, “A pure prime product version of the Hilbert Nullstellensatz”, Mich. Math. J. 45 (1998), p. 581-597 Article | MR 1653279 | Zbl 0964.14002
[14] J-P Demailly, Complex Analytic and Differential Geometry, monograph, Grenoble, 1997
[15] A. Dickenstein & C. Sessa, “Canonical representatives in moderate cohomology”, Invent. Math. 80 (1985), p. 417-434 Article | MR 791667 | Zbl 0556.32005
[16] L. Ein & R. Lazarsfeld, “A geometric effective Nullstellensatz”, Invent. math. 135 (1999), p. 427-448 Article | MR 1705839 | Zbl 0944.14003
[17] A. Fabiano, A. Ploski & P. Tworzewski, “Effective Nullstellensatz for strictly regular sequences”, Univ. Iagel. Acta Math. 38 (2000), p. 163-167 MR 1812110 | Zbl 01797615
[18] P. Griffiths & J. Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978 MR 507725 | Zbl 0408.14001
[19] M. Hickel, “Solution d’une conjecture de C. Berenstein-A. Yger et invariants de contact à l’infini”, Ann. Inst. Fourier 51 (2001), p. 707-744 Cedram | MR 1838463 | Zbl 0991.13009
[20] Z. Jelonek, “On the effective Nullstellensatz”, preprint, 2004 Zbl 1087.14003
[21] J. Kollár, “Sharp effective Nullstellensatz”, J. American Math. Soc. 1 (1988), p. 963-975 Article | MR 944576 | Zbl 0682.14001
[22] F.S. Macaulay, The algebraic theory of modular systems, Cambridge Univ. Press, Cambridge, 1916 MR 1281612 | Zbl 0802.13001 | JFM 46.0167.01
[23] M. Nöther, “Über einen Satz aus der Theorie der algebraischen Functionen”, Math. Ann. (1873), p. 351-359 Article | MR 1509826 | JFM 05.0348.02
[24] M. Passare, “Residues, currents, and their relation to ideals of holomorphic functions”, Math. Scand. 62 (1988), p. 75-152 MR 961584 | Zbl 0633.32005
[25] M. Passare, A. Tsikh & A. Yger, “Residue currents of the Bochner-Martinelli type”, Publ. Mat. 44 (2000), p. 85-117 MR 1775747 | Zbl 0964.32003
[26] B. Shiffman, “Degree bounds for the division problem in polynomial ideals”, Michigan Math. J. 36 (1989), p. 163-171 Article | MR 1000520 | Zbl 0691.12010
[27] H. Skoda, “Application des techniques $L^{2}$ à la théorie des idéaux d’une algèbre de fonctions holomorphes avec poids”, Ann. Sci. École Norm. Sup. 5 (1972), p. 545-579 Numdam | MR 333246 | Zbl 0254.32017
[28] B. Teissier, “Résultats récents d’algèbre commutative effective”, Séminaire Bourbaki 1989/90 (1990) Numdam | Zbl 0743.13017
[29] A. Tsikh, “Multidimensional residues and their applications”, Transl. Amer. Math. Soc. 103 (1992) MR 1181199 | Zbl 0758.32001
[30] A. Tsikh & A. Yger, “Residue currents. Complex analysis”, J. Math. Sci. (N. Y.) 120 (2004), p. 1916-1971 Article | MR 2085502 | Zbl 02170265
[31] A. Vidras & A. Yger, “On some generalizations of Jacobi’s residue formula”, Ann. Sci. École Norm. Sup. 34 (2001), p. 131-157 Numdam | MR 1833092 | Zbl 0991.32003
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