Geodesic
On the integral criteria for a convergence of multidimensional Dirichlet series
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 11 (2014), pp. 76-86.

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

We consider Dirichlet series associated with a set of $m$ polynomials in $n$ variables. Such series depend on $m$ complex parameters. They were studed by B. Lichtin and others in the case of hypoelliptic polynomials. We consider a more general class of polynomials so called quasielliptic polynomials in the sence of T. Ermolaeva and A. Tsikh. Using the toric geometry we discribe the domain of convergence in terms of Newton polytopes of polynomials defining the series. As an auxiliary statement we give a criterion for convergence of some integrals over $\mathbb {R}^n$.
Keywords: multidimensional Dirichlet series, Newton polytope, toric variaty.
Mots-clés : quasi-elliptic polinomial 
@article{SEMR_2014_11_a76,
     author = {E. V. Zubchenkova},
     title = {On the integral criteria for a convergence of multidimensional {Dirichlet} series},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {76--86},
     publisher = {mathdoc},
     volume = {11},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2014_11_a76/}
}
TY  - JOUR
AU  - E. V. Zubchenkova
TI  - On the integral criteria for a convergence of multidimensional Dirichlet series
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2014
SP  - 76
EP  - 86
VL  - 11
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2014_11_a76/
LA  - ru
ID  - SEMR_2014_11_a76
ER  - 
%0 Journal Article
%A E. V. Zubchenkova
%T On the integral criteria for a convergence of multidimensional Dirichlet series
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2014
%P 76-86
%V 11
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2014_11_a76/
%G ru
%F SEMR_2014_11_a76
E. V. Zubchenkova. On the integral criteria for a convergence of multidimensional Dirichlet series. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 11 (2014), pp. 76-86. http://geodesic.mathdoc.fr/item/SEMR_2014_11_a76/

[1] H. Mellin, “Die Dirichlet'schen Reihen, die zahlentheoretischen Funktionen und die unendlichen Produkte von endlichem Geschlecht”, Acta Math., 28 (1904), 37–64 | DOI | MR | Zbl

[2] B. Lichtin, “The asymptotics of a lattice point problem associated to a finite number of polynomials, I”, Duke Math. J., 63:1 (1991), 139–192 | DOI | MR | Zbl

[3] B. Lichtin, “The asymptotics of a lattice point problem associated to a finite number of polynomials, II”, Duke Math. J., 77:3 (1995), 699–751 | DOI | MR | Zbl

[4] T. O. Ermolaeva, A. K. Tsikh, “Integrirovanie ratsionalnykh funktsii po $\mathbb{R}^n$ s pomoschyu toricheskikh kompaktifikatsii i mnogomernykh vychetov”, Matem. sbornik, 187:9 (1996), 45–64 | DOI | MR | Zbl

[5] E. V. Zubchenkova, “Integralnyi priznak skhodimosti nekotorykh kratnykh ryadov”, Journal of Siberian Federal University. Mathematics, 4:3 (2011), 344–349

[6] A. K. Tsikh, “Usloviya absolyutnoi skhodimosti ryada iz koeffitsientov Teilora meromorfnykh funktsii dvukh peremennykh”, Matem. sbornik, 182:11 (1991), 1588–1622 | MR

[7] A. G. Khovanskii, “Mnogogranniki Nyutona (razreshenie osobennostei)”, Itogi nauki i tekhniki. Sovr. probl. matem., 22, 1983, 207–239 | MR

[8] C. Berkesch, J. Forsgard, M. Passare, Euler–Mellin integrals and A-hypergeometric functions, 1 Feb. 2013, arXiv: 1103.6273v2 [math.CV]