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Zeta functions for analytic mappings, log-principalization of ideals, and Newton polyhedra


Authors: Willem Veys and W. A. Zúñiga-Galindo
Journal: Trans. Amer. Math. Soc. 360 (2008), 2205-2227
MSC (2000): Primary 11S40, 11D79, 14M25; Secondary 32S45
DOI: https://doi.org/10.1090/S0002-9947-07-04422-4
Published electronically: November 28, 2007
MathSciNet review: 2366980
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Abstract: In this paper we provide a geometric description of the possible poles of the Igusa local zeta function $ Z_{\Phi }(s,\mathbf{f}) $ associated to an analytic mapping $ \mathbf{f}=$ $ \left(f_{1},\ldots ,f_{l}\right) :U(\subseteq K^{n})\rightarrow K^{l} $, and a locally constant function $ \Phi$, with support in $ U $, in terms of a log-principalizaton of the $ K\left[x \right] - $ideal $ \mathcal{I}_{\mathbf{f}}=\left(f_{1},\ldots ,f_{l}\right)$. Typically our new method provides a much shorter list of possible poles compared with the previous methods. We determine the largest real part of the poles of the Igusa zeta function, and then as a corollary, we obtain an asymptotic estimation for the number of solutions of an arbitrary system of polynomial congruences in terms of the log-canonical threshold of the subscheme given by $ \mathcal{I}_{\mathbf{f}} $. We associate to an analytic mapping $ \boldsymbol{f} $ $ = $ $ \left(f_{1},\ldots ,f_{l}\right) $ a Newton polyhedron $ \Gamma \left(\boldsymbol{f}\right) $ and a new notion of non-degeneracy with respect to $ \Gamma \left(\boldsymbol{f}\right) $. The novelty of this notion resides in the fact that it depends on one Newton polyhedron, and Khovanskii's non-degeneracy notion depends on the Newton polyhedra of $ f_{1},\ldots ,f_{l} $ . By constructing a log-principalization, we give an explicit list for the possible poles of $ Z_{\Phi }(s,\mathbf{f}) $, $ l\geq 1 $, in the case in which $ \mathbf{f} $ is non-degenerate with respect to $ \Gamma \left(\boldsymbol{f}\right) $.


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  • 1. Artal Bartolo E., Cassou-Noguès P., Luengo I., and Melle Hernández A., Quasi-ordinary power series and their zeta functions, Mem. Amer. Math. Soc., vol. 178, no. 841 (2005). MR 2172403 (2007d:14005)
  • 2. Arnold V. I., Gussein-Zade S. M., Varchenko A. N., Singularités des applications différentiables, Vol II, Éditions Mir, Moscou, 1986.
  • 3. Denef J., Report on Igusa's Local Zeta Function, Séminaire Bourbaki 43 (1990-1991), exp. 741; Astérisque 201-202-203 (1991), 359-386. Available at http://www.wis.kuleuven.ac.be/algebra/denef.html. MR 1157848 (93g:11119)
  • 4. Denef J., The rationality of the Poincaré series associated to the $ p-$adic points on a variety, Invent. Math. 77 (1984), 1-23. MR 751129 (86c:11043)
  • 5. Denef J., Poles of $ p$-adic complex powers and Newton polyhedra, Nieuw. Arch. Wisk. 13 (1995), 289-295. MR 1378800 (96m:11106)
  • 6. Denef J., On the degree of Igusa's local zeta function, Amer. J. Math. 109 (1987), 991-1008. MR 919001 (89d:11108)
  • 7. Denef J. and Hoornaert K., Newton polyhedra and Igusa's local zeta function, J. Number Theory 89 (2001), 31-64. MR 1838703 (2002g:11170)
  • 8. Denef J. and Loeser F., Motivic Igusa zeta functions, J. Alg. Geom. 7 (1998), 505-537. MR 1618144 (99j:14021)
  • 9. Denef J. and Loeser F., Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math. 135 (1999), 201-232. MR 1664700 (99k:14002)
  • 10. Denef J. and Loeser F., Caractéristiques d'Euler-Poincaré, fonctions zêta locales et modifications analytiques, J. Amer. Math. Soc. 5 (1992), 705-720. MR 1151541 (93g:11118)
  • 11. Encinas S., Nobile A., and Villamayor O., On algorithmic equi-resolution and stratification of Hilbert schemes, Proc. London Math. Soc. 86 (2003), no. 3, 607-648. MR 1974392 (2004e:14027)
  • 12. Hironaka H., Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. Math. 79 (1964), 109-326. MR 0199184 (33:7333)
  • 13. Howald J., Mustata M., and Yuen C. O., On Igusa zeta functions of monomial ideals. Available at http://www.arxiv.org/abs/math.AG/0509243.
  • 14. Igusa J.-I., Complex powers of irreducible algebroid curves, Geometry Today, Birkhäuser (1985), 207-230. MR 895155 (88j:11084)
  • 15. Igusa J.-I., Some aspects of the arithmetic theory of polynomials. Discrete groups in geometry and analysis (New Haven, Conn., 1984), 20-47, Progr. Math., 67, Birkhäuser Boston, Boston, MA, 1987. MR 900822 (89e:11074)
  • 16. Igusa J.-I., An introduction to the theory of local zeta functions, AMS/IP Studies in Advanced Mathematics, 2000. MR 1743467 (2001j:11112)
  • 17. Kempf G., Knudsen F., Mumford D., Saint-Donat B., Toroidal embeddings, Lectures Notes in Mathematics vol. 339, Springer-Verlag, 1973. MR 0335518 (49:299)
  • 18. Khovanskii A. G., Newton polyhedra and toroidal varieties, Funct. Anal. Appl. 12 (1978), no.1, 51-61. MR 487230 (80b:14022)
  • 19. Kollár J., Singularities of pairs, Summer Research Institute on Algebraic Geometry (Santa Cruz 1995), Amer. Math. Soc., Proc. Symp. Pure Math. 62, Part 1 (1997), 221-287. MR 1492525 (99m:14033)
  • 20. Langlands R. P., Orbital integrals on forms of SL(3). I, Amer. J. Math., 105 (1983), 465-506. MR 701566 (86d:22012)
  • 21. Lichtin B., Meuser D., Poles of local zeta functions and Newton polygons, Compositio Math. 55 (1985), 313-332. MR 799820 (87a:11120)
  • 22. Loeser F., Une estimation asymptotique du nombre de solutions approchées d'une équation $ p$-adique, Invent. Math. 85 (1986), no. 1, 31-38. MR 842046 (87j:11136)
  • 23. Meuser D., On the poles of a local zeta function for curves, Invent. Math. 73 (1983), 445-465. MR 718941 (85i:14014)
  • 24. Meuser D., On the rationality of certain generating functions, Math. Ann. 256 (1981), 303-310. MR 626951 (83g:12015)
  • 25. Mustata M., Singularities of pairs via jet schemes. J. Amer. Math. Soc. 15 (2002), no. 3, 599-615. MR 1896234 (2003b:14005)
  • 26. Oka M., Principal zeta-function of non-degenerate complete intersection singularity, J. Fac. Sci. Tokyo Sect. IA, Math. 37 (1990), 11-32. MR 1049017 (92b:32041)
  • 27. Strauss L., Poles of two-variable $ p-$adic complex power, Trans. Amer. Math. Soc., 27 (1983), 481-493. MR 701506 (84k:14019)
  • 28. Saia M.J., The integral closure of ideals and the Newton filtration, J. Alg. Geom. 5 (1996), 1-11. MR 1358033 (96m:14003)
  • 29. Saia M.J. and Zuniga-Galindo W.A., Local zeta functions for curves, non-degeneracy conditions and Newton polygons, Trans. Amer. Math. Soc., 357 (2005), 59-88. MR 2098087 (2006b:11150)
  • 30. Teissier B., Variétés polaires. I. Invariants polaires des singularités d'hypersurfaces, Invent. Math. 40 (1977), no. 3, 267-292. MR 0470246 (57:10004)
  • 31. Varchenko A., Newton polyhedra and estimation of oscillating integrals, Funct. Anal. Appl. 10 (1976), 175-196.
  • 32. Veys W., Arc spaces, motivic integration and stringy invariants, Advanced Studies in Pure Mathematics, Proceedings of Singularity Theory and its applications, Sapporo (Japan), 16-25 September 2003 (to appear), 43p.
  • 33. Veys W., On the poles of Igusa's local zeta functions for curves, J. London Math. Soc., 41 (1990), 27-32. MR 1063539 (92j:11142)
  • 34. Veys W., Poles of Igusa's local zeta function and monodromy, Bull. Soc. Math. Fr., 121 (1993), 545-598. MR 1254752 (95b:11110)
  • 35. W\lodarczyk J., Simple Hironaka resolution in characteristic zero, J. Amer. Math. Soc. 18 (2005), 779-822. MR 2163383 (2006f:14014)
  • 36. Zuniga-Galindo W.A., Local zeta functions and Newton polyhedra, Nagoya Math. J. 172 (2003), 31-58. MR 2019519 (2004h:11098)
  • 37. Zuniga-Galindo W.A., On the poles of Igusa's local zeta function for algebraic sets, Bull. London Math. Soc. 36 (2004), 310-320. MR 2038719 (2005c:11149)
  • 38. Zuniga-Galindo W.A., Local zeta function for non-degenerate homogeneous mappings, Pacific J. Math. 1 (2005), 187-200. MR 2224596 (2007a:11160)

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Additional Information

Willem Veys
Affiliation: Department of Mathematics, University of Leuven, Celestijnenlaan 200 B, B-3001 Leuven (Heverlee), Belgium
Email: wim.veys@wis.kuleuven.be

W. A. Zúñiga-Galindo
Affiliation: Department of Mathematics and Computer Science, Barry University, 11300 N.E. Second Avenue, Miami Shores, Florida 33161
Address at time of publication: Departamento de Matemáticas, Centro de Investigacion y Estudios Avanzados del I.P.N., Av. Inst. Politécnico Nacional 2508, C.P. 07360, México D.F., México
Email: wzuniga@mail.barry.edu, wzuniga@math.cinvestav.mx

DOI: https://doi.org/10.1090/S0002-9947-07-04422-4
Keywords: Igusa zeta functions, congruences in many variables, topological zeta functions, motivic zeta functions, Newton polyhedra, toric varieties, log-principalization of ideals
Received by editor(s): January 9, 2006
Received by editor(s) in revised form: September 1, 2006
Published electronically: November 28, 2007
Additional Notes: The first author was partially supported by the Fund of Scientific Research – Flanders (G.0318.06).
The second author thanks the financial support of the NSA. Project sponsored by the National Security Agency under Grant Number H98230-06-1-0040. The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation herein.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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