Hatziafratis, Telemachos On the multiplicity of holomorphic maps and a residue formula. (English) Zbl 0804.32004 Rend. Semin. Mat. Univ. Padova 89, 37-45 (1993). The (algebraic or geometric) multiplicity of holomorphic map \((f_ 1, \dots, f_ n)\) at an isolated zero \(0 \in \mathbb{C}^ n\) can be expressed as the local residue at 0 of \({df_ 1 \over f_ 1} \wedge \cdots \wedge {df_ n \over f_ n}\), which by means of the Bochner-Martinelli kernel can be written as an integral over a small \((2n-1)\)-ball around the origin. Under appropriate hypotheses and based on Stokes theorem, the author presents integral formulas over \((2n-2p-1)\) dimensional cycles (for \(1 \leq p \leq n-1)\) for the computation of this local multiplicity. Reviewer: A.Dickenstein (Buenos Aires) MSC: 32A27 Residues for several complex variables 32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.) Keywords:residue; local multiplicity PDFBibTeX XMLCite \textit{T. Hatziafratis}, Rend. Semin. Mat. Univ. Padova 89, 37--45 (1993; Zbl 0804.32004) Full Text: Numdam EuDML References: [1] I. Aizenberg - A. Yuzhakov , Integral representations and residues in multidimensional complex analysis , Transl. Math. Monos. , 58 , Amer. Math. Soc. , Providence, R.I. ( 1983 ). MR 735793 | Zbl 0537.32002 · Zbl 0537.32002 [2] M. Berger - B. GOSTIAUX, Differential Geometry Manifolds, Curves and Surfaces , Springer-Verlag , New York ( 1988 ). MR 917479 | Zbl 0629.53001 · Zbl 0629.53001 [3] P. Griffiths - J. Harris , Principles of Algebraic Geometry , Wiley , New York ( 1978 ). MR 507725 | Zbl 0836.14001 · Zbl 0836.14001 [4] T. Hatziafratis , On certain integrals associated to CR-functions , Trans. Amer. Math. Soc. , 314 ( 2 ) ( 1989 ), pp. 781 - 802 . MR 958894 | Zbl 0682.32002 · Zbl 0682.32002 · doi:10.2307/2001408 [5] T. Hatziafratis , A global residue theorem on analytic varieties , J. Math. Anal. Appl. , 149 ( 2 ) ( 1990 ), pp. 475 - 488 . MR 1057688 | Zbl 0712.32004 · Zbl 0712.32004 · doi:10.1016/0022-247X(90)90056-L [6] J. Milnor , Singular points of complex hypersurfaces , Ann. Math. Studies , 61 , Princeton Univ. Press , New Jersey ( 1968 ). MR 239612 | Zbl 0184.48405 · Zbl 0184.48405 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.