Monodromy zeta-function of a polynomial on a complete intersection, and Newton polyhedra

Gusev, G.

doi:10.1090/S1061-0022-2012-01205-7

Monodromy zeta-function of a polynomial on a complete intersection, and Newton polyhedra
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by G. G. Gusev
Translated by: the author

St. Petersburg Math. J. 23 (2012), 511-519

DOI: https://doi.org/10.1090/S1061-0022-2012-01205-7

Published electronically: March 2, 2012

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Abstract:

For a generic (polynomial) one-parameter deformation of a complete intersection, its monodromy zeta-function is defined. Explicit formulas for this zeta-function in terms of the corresponding Newton polyhedra are obtained in the case where the deformation is nondegenerate with respect to its Newton polyhedra. This result is employed to obtain a formula for the monodromy zeta-function at the origin of a polynomial on a complete intersection, which is an analog of the Libgober–Sperber theorem.

References

S. M. Guseĭn-Zade, I. Luengo, and A. Mel′e-Èrnadez, Zeta functions of germs of meromorphic functions, and the Newton diagram, Funktsional. Anal. i Prilozhen. 32 (1998), no. 2, 26–35, 95 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 32 (1998), no. 2, 93–99. MR 1647824, DOI 10.1007/BF02482595
S. M. Guseĭn-Zade and D. Sirsma, Deformations of polynomials and their zeta functions, Sovrem. Mat. Prilozh. 33 (2005), 36–42 (Russian, with Russian summary); English transl., J. Math. Sci. (N.Y.) 144 (2007), no. 1, 3782–3788. MR 2464745, DOI 10.1007/s10958-007-0231-1
Gleb Gusev, Monodromy zeta-functions of deformations and Newton diagrams, Rev. Mat. Complut. 22 (2009), no. 2, 447–454. MR 2553942, DOI 10.5209/rev_{R}EMA.2009.v22.n2.16282
A. Libgober and S. Sperber, On the zeta function of monodromy of a polynomial map, Compositio Math. 95 (1995), no. 3, 287–307. MR 1318089
Yutaka Matsui and Kiyoshi Takeuchi, Monodromy zeta functions at infinity, Newton polyhedra and constructible sheaves, Math. Z. 268 (2011), no. 1-2, 409–439. MR 2805442, DOI 10.1007/s00209-010-0678-5
Dirk Siersma and Mihai Tibăr, Deformations of polynomials, boundary singularities and monodromy, Mosc. Math. J. 3 (2003), no. 2, 661–679, 745 (English, with English and Russian summaries). Dedicated to Vladimir I. Arnold on the occasion of his 65th birthday. MR 2025278, DOI 10.17323/1609-4514-2003-3-2-661-679
O. Ya. Viro, Some integral calculus based on Euler characteristic, Topology and geometry—Rohlin Seminar, Lecture Notes in Math., vol. 1346, Springer, Berlin, 1988, pp. 127–138. MR 970076, DOI 10.1007/BFb0082775
G. Buzeman, Vypuklye poverkhnosti, Izdat. “Nauka”, Moscow, 1964 (Russian). MR 0178403
A. G. Hovanskiĭ, Newton polyhedra, and toroidal varieties, Funkcional. Anal. i Priložen. 11 (1977), no. 4, 56–64, 96 (Russian). MR 0476733
A. G. Hovanskiĭ, Newton polyhedra, and the genus of complete intersections, Funktsional. Anal. i Prilozhen. 12 (1978), no. 1, 51–61 (Russian). MR 487230
A. N. Varčenko, Theorems on the topological equisingularity of families of algebraic varieties and families of polynomial mappings, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 957–1019 (Russian). MR 0337956