Geodesic
On computability of multiple integrals by means of a sum of local residues
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 15 (2018), pp. 996-1010.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider $n$-fold integrals of meromorphic differential $n$-forms on an $n$-dimensional complex manifold and study the problem of computability of such integrals by means of local (Grothendieck) residues of these forms. This problem is relevant in various fields of theoretical physics (in superstring theory for study of periods of Calabi–Yau manifolds, in particle physics for computation of anomalous magnetic moments of muons). The obtained theorems refine earlier results of A.K. Tsikh and A.P. Yuzhakov.
Keywords: local residue, local cycle, separating cycle. 
@article{SEMR_2018_15_a133,
     author = {R. V. Ulvert},
     title = {On computability of multiple integrals by means of a sum of local residues},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {996--1010},
     publisher = {mathdoc},
     volume = {15},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2018_15_a133/}
}
TY  - JOUR
AU  - R. V. Ulvert
TI  - On computability of multiple integrals by means of a sum of local residues
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2018
SP  - 996
EP  - 1010
VL  - 15
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2018_15_a133/
LA  - ru
ID  - SEMR_2018_15_a133
ER  - 
%0 Journal Article
%A R. V. Ulvert
%T On computability of multiple integrals by means of a sum of local residues
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2018
%P 996-1010
%V 15
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2018_15_a133/
%G ru
%F SEMR_2018_15_a133
R. V. Ulvert. On computability of multiple integrals by means of a sum of local residues. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 15 (2018), pp. 996-1010. http://geodesic.mathdoc.fr/item/SEMR_2018_15_a133/

[1] A.K. Tsikh, Multidimensional Residues and their Applications, Translations of Mathematical Monographs, 103, AMS, Providence, 1992 | DOI | MR | Zbl

[2] O.N. Zhdanov, A.K. Tsikh, “Studying the multiple Mellin–Barnes integrals by means of multidimensional residues”, Siberian Mathematical Journal, 39:2 (1998), 245–260 | DOI | MR | Zbl

[3] S. Friot, D. Greynat, “On convergent series representation of Mellin–Barnes integrals”, Journal of Mathematical Physics, 53:2 (2012), 023508 | DOI | MR | Zbl

[4] M. Passare, A.K. Tsikh, A.A. Cheshel, “Multiple Mellin–Barnes integrals as periods of Calabi–Yau manifolds with several moduli”, Theoretical and Mathematical Physics, 109:3 (1997), 1544–1555 | DOI | MR

[5] J. Charles, D. Greynat, E. de Rafael, “The Mellin–Barnes approach to hadronic vacuum polarization and $g_{\mu}-2$”, Physical Review D, 97 (2018), 076014 | DOI

[6] K.V. Safonov, A.K. Tsikh, “Singularities of the Grothendieck parametric residue and diagonals of a double power series”, Soviet Mathematics (Izvestiya VUZ. Matematika), 28:4 (1984), 65–74 | MR | Zbl

[7] D.Yu. Pochekutov, “Diagonals of the Laurent series of rational functions”, Siberian Mathematical Journal, 50:6 (2009), 1081–1091 | DOI | MR | Zbl

[8] O.I. Egorushkin, I.V. Kolbasina, K.V. Safonov, “On application of multidimensional complex analysis in formal language and grammar theory”, Applied Discrete Mathematics, 37 (2017), 76–89 (in Russian) | DOI | MR

[9] P. Griffiths, J. Harris, Principles of algebraic geometry, John Wiley and Sons, New York, 1978 | MR | Zbl

[10] A.K. Tsikh, “Cycles separating zeros of analytic functions in $\mathbb{C}^n$”, Siberian Mathematical Journal, 16:5 (1975), 859–862 | DOI | MR

[11] A.P. Yuzhakov, “A coboundary condition of Leray and its application to logarithmic residues”, Siberian Mathematical Journal, 11:3 (1970), 540–542 | DOI | MR | Zbl

[12] A.P. Yuzhakov, “A separating subgroup and local residues”, Siberian Mathematical Journal, 29:6 (1988), 1028–1033 | DOI | MR | Zbl

[13] R.V. Ulvert, “Homological Resolutions in Problems About Separating Cycles”, Siberian Mathematical Journal, 59:3 (2018), 542–550 | DOI | Zbl

[14] A.P. Yuzhakov, “On the separation of analytic singularities and the decomposition of holomorphic functions of n variables into partial fractions”, Multidimensional complex analysis, IF SO USSR, Krasnoyarsk, 1986, 210–220 (in Russian) | MR

[15] E.K. Leinartas, “Factorization of rational functions of several variables into partial fractions”, Soviet Mathematics (Izvestiya VUZ. Matematika), 22:10 (1978), 35–38 | MR