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


Author: G. G. Gusev
Translated by: the author
Original publication: Algebra i Analiz, tom 23 (2011), nomer 3.
Journal: St. Petersburg Math. J. 23 (2012), 511-519
MSC (2010): Primary 14Q15, 14D05; Secondary 58K15, 58K10, 32S20
DOI: https://doi.org/10.1090/S1061-0022-2012-01205-7
Published electronically: March 2, 2012
MathSciNet review: 2896166
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Abstract | References | Similar Articles | Additional Information

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.


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  • 1. S. M. Guseĭn-Zade, I. Luengo, and A. Melle-Hernandez, Zeta functions of germs of meromorphic functions and Newton diagram, Funktsional. Anal. i Prilozhen. 32 (1998), no. 2, 26-35; English transl., Funct. Anal. Appl. 32 (1998), no. 2, 93-99. MR 1647824 (99j:32040)
  • 2. S. M. Guseĭn-Zade and D. Sirsma (Siersma), Deformation of polynomials and their zeta functions, Itogi Nauki i Tekhniki Sovrem. Mat. i Prilozhen., No. 33, VINITI, Moscow, 2005, pp. 36-42; English transl., J. Math. Sci. (N. Y.) 144 (2007), no. 1, 3782-3788. MR 2464745 (2010b:32042)
  • 3. G. G. Gusev, Monodromy zeta-functions of deformations and Newton diagrams, Rev. Mat. Complut. 22 (2009), 447-454. MR 2553942 (2011c:32047)
  • 4. A. Libgober and S. Sperber, On the zeta function of monodromy of a polynomial map, Compositio Math. 95 (1995), 287-307. MR 1318089 (96b:14022)
  • 5. Y. Matsui and K. Takeuchi, Monodromy zeta-function at infinity, Newton polyhedra, and constructible sheaves, Math. Z., 268 (2011), no. 1-2, 409-439. MR 2805442
  • 6. D. Siersma and M. Tibar, Deformations of polynomials, boundary singularities and monodromy, Mosc. Math. J. 3 (2003), 661-679. MR 2025278 (2005c:32035)
  • 7. O. Y. Viro, Some integral calculus based on Euler characteristic, Topology and Geometry -- Rohlin Seminar, Lecture Notes in Math., vol. 1346, Springer-Verlag, Berlin, 1988, pp. 127-138. MR 0970076 (90a:57029)
  • 8. G. Buzeman, Convex surfaces, Nauka, Moscow, 1964. (Russian) MR 0178403 (31:2660)
  • 9. A. G. Khovanskiĭ, Newton polyhedra and toroidal varieties, Funktsional. Anal. i Prilozhen. 11 (1977), no. 4, 56-64; English transl., Funct. Anal. Appl. 11 (1977), no. 4, 289-296 (1978). MR 0476733 (57:16291)
  • 10. -, Newton polyhedra and the genus of complete intersections, Funktsional. Anal. i Prilozhen. 12 (1978), no. 1, 51-61; English transl., Funct. Anal. Appl. 12 (1978), no. 1, 38-46. MR 0487230 (80b:14022)
  • 11. A. N. Varchenko, Theorems on the topological equisingularity of families of algebraic varieties and families of polynomial mappings, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), no. 5, 957-1019; English transl., Math. USSR-Izv. 6 (1972), 949-1008. MR 0337956 (49:2725)

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Additional Information

G. G. Gusev
Affiliation: Moscow Institute of Physics and Technology, Independent University of Moscow, Russia
Email: gusev@mccme.ru

DOI: https://doi.org/10.1090/S1061-0022-2012-01205-7
Keywords: Deformations of polynomials, monodromy zeta-function, Newton polyhedron
Received by editor(s): September 23, 2009
Published electronically: March 2, 2012
Additional Notes: Partially supported by the grants RFBR-10-01-00678, RFBR-08-01-00110-a, RFBR and SU HSE 09-01-12185-off-m, and NOSH-8462.2010.1.
Article copyright: © Copyright 2012 American Mathematical Society

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