Racines de polynômes de Bernstein
Annales de l'Institut Fourier, Volume 36 (1986) no. 4, p. 1-30

Let P be a polynomial with non negative real coefficients, in two indeterminates. One shows that the knowledge of the poles of the integrals

0101x1β1-1x2β2-1P(x1,x2)sdx1dx2

gives some of the roots of the Bernstein polynomial of P. One can calculate poles of these integrals using some Mellin’s methods. Some explicit computations are given.

On considère un polynôme P, à coefficients réels non négatifs, à deux indéterminées. On montre que la connaissance des pôles des intégrales

0101x1β1-1x2β2-1P(x1,x2)sdx1dx2

donne des renseignements sur les racines du polynômes de Bernstein de P. La détermination des pôles des intégrales peut se faire en utilisant certaines méthodes de Mellin. Des calculs explicites sont donnés.

@article{AIF_1986__36_4_1_0,
     author = {Cassou-Nogu\`es, Pierrette},
     title = {Racines de polyn\^omes de Bernstein},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {36},
     number = {4},
     year = {1986},
     pages = {1-30},
     doi = {10.5802/aif.1067},
     zbl = {0597.32004},
     mrnumber = {88c:32012},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_1986__36_4_1_0}
}
Cassou-Noguès, Pierrette. Racines de polynômes de Bernstein. Annales de l'Institut Fourier, Volume 36 (1986) no. 4, pp. 1-30. doi : 10.5802/aif.1067. http://www.numdam.org/item/AIF_1986__36_4_1_0/

[1] V. Arnold, On some problems in singularity theory, Geometry and Analysis, Papers dedicated to the memory of V. K. Patodi, Springer Verlag, 1981. | Zbl 0492.58006

[2] I. N. Bernstein, Feasibility of the analytic continuation fλ+ for certain polynomials f translated from Funktsional'nyi Analiz i Ego Prilozheniya, vol. 2, n° 1, p. 92-93, January-March 1968. | MR 37 #5690 | Zbl 0181.14904

[3] P. Cassou-Noguès, Séries de Dirichlet et intégrales associées à un polynôme à deux indéterminées, Journal of Number Theory, Vol. 23, n° 1 (1986), 1-54. | MR 87j:11086 | Zbl 0584.10022

[4] M. Kashiwara, B functions and holonomic systems, Rationality of roots of b functions, Invent. Math., (1976-1977), 33-53. | Zbl 0354.35082

[5] B. Malgrange, Le polynôme de Bernstein d'une singularité isolée, Lecture Notes in Math., vol. 459, Springer Verlag 1975, 98-119. | MR 54 #7845 | Zbl 0308.32007

[6] T. Yano, On the theory of b-functions, Publ. Res. Inst. Math. Sci., 14 (1978), 111-202. | MR 80h:32026 | Zbl 0389.32005

[7] T. Yano, b-functions and exponents of hypersurface isolated singularities, Proceedings of Symposia in Pure Mathematics, vol. 40 (1983), Part. 2. | MR 85b:32012 | Zbl 0547.32004

[8] F. Ehlers et K. Lo, Minimal characteristic exponent of the Gauss-Manin connection of isolated singular point and Newton polyhedron, Math. Ann., 259 (1982), 431-441. | MR 83j:32009 | Zbl 0469.32004