Geodesic
Local residues in $\mathbf C^n$. Algebraic applications
Sbornik. Mathematics, Tome 51 (1985) no. 1, pp. 225-237 
A. K. Tsikh. Local residues in $\mathbf C^n$. Algebraic applications. Sbornik. Mathematics, Tome 51 (1985) no. 1, pp. 225-237. http://geodesic.mathdoc.fr/item/SM_1985_51_1_a13/
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Voir la notice de l'article provenant de la source Math-Net.Ru

Connected with a ingular point $a$ of an algebraic set $V=\{z\in\mathbf C^n:g(z)=0\}$ is the local residue \begin{equation} \operatorname{res}\limits_{\Gamma_a}(f/g)=\int_{\Gamma_a}\frac{f(z)}{g(z)}\,dz, \end{equation} of the rational function $f/g$, where $\Gamma_a$ is a cycle which has a representative in the $n$-dimensional homology group $H_n(\mathbf C^n\setminus V)$ in every neighborhood of the point $a$. The structure of the local residues of the form (1) is described in the case of an isolated singular point $a$: they are expressed in terms of finitely many derivatives of $f$ at $a$. As an application of local residues a theorem of Noether and Bertini is generalized to any number of variables. Bibliography: 17 titles.

[1] Tong Y. L., “Integral representation formulae and Grotendieck residue simbol”, Amer. J. Math., 95 (1973), 904–917 | DOI | MR | Zbl

[2] Griffiths P., Harris J., Principles of algebraic geometry, A. Wiley Intersience Publication, 1978 | MR

[3] Carrel J. B., “A remark on the Grothendick residue map”, Proc. of the Amer. Math. Soc., 70:1 (1978), 43–48 | DOI | MR | Zbl

[4] Yuzhakov A. P., “O vychetakh ratsionalnykh funktsii dvukh peremennykh”, Golomorfnye funktsii mnogikh kompleksnykh peremennykh, IF SOAN SSSR, Krasnoyarsk, 1973, 181–191

[5] Aizenberg L. A., Yuzhakov A. P., Integralnye predstavleniya i vychety v mnogomernom kompleksnom analize, Nauka, Novosibirsk, 1979

[6] Mysovskikh I. P., Interpolyatsionnye kubaturnye formuly, Nauka, M., 1981 | MR | Zbl

[7] Bertini E., “Zum Fundamentalsatz aus der Teorie der algebraischen Funktionen”, Math. Ann. B, 34 (1889), 447–449 | DOI | MR

[8] Milnor Dzh., Osobye tochki kompleksnykh giperpoverkhnostei, Mir, M., 1971 | MR | Zbl

[9] Lojasiewicz S., “Triangulation of semi-analitic sets”, Ann. Scu. Norm. Sup. Pisa, Sc. Fis. Mat. Ser. 3, 18:4 (1964), 449–474 | MR | Zbl

[10] Mamford D., Algebraicheskaya geometriya, T. 1, Mir, M., 1979 | MR

[11] Khërmander L., Vvedenie v teoriyu funktsii neskolkikh kompleksnykh peremennykh, Mir, M., 1968 | MR

[12] Leinartas E. K., “Ob odnorodnosti po parametram nekotorykh integralov”, O golomorfnykh funktsiyakh mnogikh kompleksnykh peremennykh, IF SOAN SSSR, Krasnoyarsk, 1976, 190–192

[13] Odvirko-Budko B. I., “Teorema edinstvennosti P. Montelya v sluchae $n$ $(n\geq2)$ peremennykh i nekotorye ee primeneniya”, Izv. VUZov. Matematika, 6 (1979), 43–51 | Zbl

[14] Shabat B. V., Vvedenie v kompleksnyi analiz, T. 2, Nauka, M., 1976 | MR

[15] Noether M., “Über einen satz aus der Teorie der algebraischen Funktionen”, Math. Ann. B, 6 (1873), 351–359 | DOI

[16] Van der Varden B. L., Algebra, Nauka, M., 1976 | MR

[17] Yuzhakov A. P., Tsikh A. K., “O kratnosti nulya sistemy golomorfnykh funktsii”, Sib. matem. zh., 19:3 (1978), 693–697 | MR | Zbl