Zeta functions for analytic mappings, log-principalization of ideals, and Newton polyhedra
HTML articles powered by AMS MathViewer
- by Willem Veys and W. A. Zúñiga-Galindo PDF
- Trans. Amer. Math. Soc. 360 (2008), 2205-2227 Request permission
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 $\Gamma \left (\boldsymbol {f}\right )$.References
- Enrique Artal Bartolo, Pierrette Cassou-Noguès, Ignacio Luengo, and Alejandro Melle Hernández, Quasi-ordinary power series and their zeta functions, Mem. Amer. Math. Soc. 178 (2005), no. 841, vi+85. MR 2172403, DOI 10.1090/memo/0841
- Arnold V. I., Gussein-Zade S. M., Varchenko A. N., Singularités des applications différentiables, Vol II, Éditions Mir, Moscou, 1986.
- Jan Denef, Report on Igusa’s local zeta function, Astérisque 201-203 (1991), Exp. No. 741, 359–386 (1992). Séminaire Bourbaki, Vol. 1990/91. MR 1157848
- J. Denef, The rationality of the Poincaré series associated to the $p$-adic points on a variety, Invent. Math. 77 (1984), no. 1, 1–23. MR 751129, DOI 10.1007/BF01389133
- J. Denef, Poles of $p$-adic complex powers and Newton polyhedra, Nieuw Arch. Wisk. (4) 13 (1995), no. 3, 289–295. MR 1378800
- J. Denef, On the degree of Igusa’s local zeta function, Amer. J. Math. 109 (1987), no. 6, 991–1008. MR 919001, DOI 10.2307/2374583
- Jan Denef and Kathleen Hoornaert, Newton polyhedra and Igusa’s local zeta function, J. Number Theory 89 (2001), no. 1, 31–64. MR 1838703, DOI 10.1006/jnth.2000.2606
- Jan Denef and François Loeser, Motivic Igusa zeta functions, J. Algebraic Geom. 7 (1998), no. 3, 505–537. MR 1618144
- Jan Denef and François Loeser, Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math. 135 (1999), no. 1, 201–232. MR 1664700, DOI 10.1007/s002220050284
- J. Denef and F. Loeser, Caractéristiques d’Euler-Poincaré, fonctions zêta locales et modifications analytiques, J. Amer. Math. Soc. 5 (1992), no. 4, 705–720 (French). MR 1151541, DOI 10.1090/S0894-0347-1992-1151541-7
- S. Encinas, A. Nobile, and O. Villamayor U., On algorithmic equi-resolution and stratification of Hilbert schemes, Proc. London Math. Soc. (3) 86 (2003), no. 3, 607–648. MR 1974392, DOI 10.1112/S0024611502013862
- Heisuke Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203; ibid. (2) 79 (1964), 205–326. MR 0199184, DOI 10.2307/1970547
- Howald J., Mustaţă M., and Yuen C. O., On Igusa zeta functions of monomial ideals. Available at http://www.arxiv.org/abs/math.AG/0509243.
- Jun-ichi Igusa, Complex powers of irreducible algebroid curves, Geometry today (Rome, 1984) Progr. Math., vol. 60, Birkhäuser Boston, Boston, MA, 1985, pp. 207–230. MR 895155
- Jun-ichi Igusa, Some aspects of the arithmetic theory of polynomials, Discrete groups in geometry and analysis (New Haven, Conn., 1984) Progr. Math., vol. 67, Birkhäuser Boston, Boston, MA, 1987, pp. 20–47. MR 900822, DOI 10.1007/978-1-4899-6664-3_{2}
- Jun-ichi Igusa, An introduction to the theory of local zeta functions, AMS/IP Studies in Advanced Mathematics, vol. 14, American Mathematical Society, Providence, RI; International Press, Cambridge, MA, 2000. MR 1743467, DOI 10.1090/amsip/014
- G. Kempf, Finn Faye Knudsen, D. Mumford, and B. Saint-Donat, Toroidal embeddings. I, Lecture Notes in Mathematics, Vol. 339, Springer-Verlag, Berlin-New York, 1973. MR 0335518, DOI 10.1007/BFb0070318
- A. G. Hovanskiĭ, Newton polyhedra, and the genus of complete intersections, Funktsional. Anal. i Prilozhen. 12 (1978), no. 1, 51–61 (Russian). MR 487230
- János Kollár, Singularities of pairs, Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 221–287. MR 1492525
- R. P. Langlands, Orbital integrals on forms of $\textrm {SL}(3)$. I, Amer. J. Math. 105 (1983), no. 2, 465–506. MR 701566, DOI 10.2307/2374265
- Ben Lichtin and Diane Meuser, Poles of a local zeta function and Newton polygons, Compositio Math. 55 (1985), no. 3, 313–332. MR 799820
- F. Loeser, Une estimation asymptotique du nombre de solutions approchées d’une équation $p$-adique, Invent. Math. 85 (1986), no. 1, 31–38 (French). MR 842046, DOI 10.1007/BF01388790
- D. Meuser, On the poles of a local zeta function for curves, Invent. Math. 73 (1983), no. 3, 445–465. MR 718941, DOI 10.1007/BF01388439
- Diane Meuser, On the rationality of certain generating functions, Math. Ann. 256 (1981), no. 3, 303–310. MR 626951, DOI 10.1007/BF01679699
- Mircea Mustaţǎ, Singularities of pairs via jet schemes, J. Amer. Math. Soc. 15 (2002), no. 3, 599–615. MR 1896234, DOI 10.1090/S0894-0347-02-00391-0
- Mutsuo Oka, Principal zeta-function of nondegenerate complete intersection singularity, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 37 (1990), no. 1, 11–32. MR 1049017
- Leon Strauss, Poles of a two-variable $P$-adic complex power, Trans. Amer. Math. Soc. 278 (1983), no. 2, 481–493. MR 701506, DOI 10.1090/S0002-9947-1983-0701506-2
- Marcelo J. Saia, The integral closure of ideals and the Newton filtration, J. Algebraic Geom. 5 (1996), no. 1, 1–11. MR 1358033
- M. J. Saia and W. A. Zuniga-Galindo, Local zeta function for curves, non-degeneracy conditions and Newton polygons, Trans. Amer. Math. Soc. 357 (2005), no. 1, 59–88. MR 2098087, DOI 10.1090/S0002-9947-03-03491-3
- B. Teissier, Variétés polaires. I. Invariants polaires des singularités d’hypersurfaces, Invent. Math. 40 (1977), no. 3, 267–292 (French). MR 470246, DOI 10.1007/BF01425742
- Varchenko A., Newton polyhedra and estimation of oscillating integrals, Funct. Anal. Appl. 10 (1976), 175-196.
- 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.
- W. Veys, On the poles of Igusa’s local zeta function for curves, J. London Math. Soc. (2) 41 (1990), no. 1, 27–32. MR 1063539, DOI 10.1112/jlms/s2-41.1.27
- Willem Veys, Poles of Igusa’s local zeta function and monodromy, Bull. Soc. Math. France 121 (1993), no. 4, 545–598 (English, with English and French summaries). MR 1254752, DOI 10.24033/bsmf.2219
- Jarosław Włodarczyk, Simple Hironaka resolution in characteristic zero, J. Amer. Math. Soc. 18 (2005), no. 4, 779–822. MR 2163383, DOI 10.1090/S0894-0347-05-00493-5
- W. A. Zuniga-Galindo, Local zeta functions and Newton polyhedra, Nagoya Math. J. 172 (2003), 31–58. MR 2019519, DOI 10.1017/S0027763000008631
- W. A. Zuniga-Galindo, On the poles of Igusa’s local zeta function for algebraic sets, Bull. London Math. Soc. 36 (2004), no. 3, 310–320. MR 2038719, DOI 10.1112/S0024609303002947
- W. A. Zuniga-Galindo, Local zeta function for nondegenerate homogeneous mappings, Pacific J. Math. 218 (2005), no. 1, 187–200. MR 2224596, DOI 10.2140/pjm.2005.218.187
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
- 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. - © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2366980