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Transactions of the American Mathematical Society

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Local zeta function for curves, non-degeneracy conditions and Newton polygons

Authors: M. J. Saia and W. A. Zuniga-Galindo
Translated by:
Journal: Trans. Amer. Math. Soc. 357 (2005), 59-88
MSC (2000): Primary 11D79, 14G20, 14M25
Published electronically: December 15, 2003
MathSciNet review: 2098087
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Abstract: This paper is dedicated to a description of the poles of the Igusa local zeta function $Z(s,f,v)$ when $f(x,y)$ satisfies a new non-degeneracy condition called arithmetic non-degeneracy. More precisely, we attach to each polynomial $f(x,y)$ a collection of convex sets $\Gamma ^{A}(f)=\left\{ \Gamma _{f,1},\dots ,\Gamma _{f,l_{0}}\right\} $called the arithmetic Newton polygon of $f(x,y)$, and introduce the notion of arithmetic non-degeneracy with respect to $\Gamma ^{A}(f)$. If $L_{v}$ is a $p$-adic field, and $f(x,y)\in L_{v}\left[ x,y \right] $ is arithmetically non-degenerate, then the poles of $Z(s,f,v)$ can be described explicitly in terms of the equations of the straight segments that form the boundaries of the convex sets $\Gamma _{f,1},\dots , \Gamma _{f,l_{0}}$. Moreover, the proof of the main result gives an effective procedure for computing $Z(s,f,v)$.

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  • 1. Arnold V., Varchenko A. and Gussein-Zade S., Singularités des applications différentiables, vol. 2, Mir, Moscow, 1986.
  • 2. Denef J., Report on Igusa's local zeta function, Seminaire Bourbaki 1990/1991 (730-744) in Asterisque 201-203 (1991), 359-386. MR 93g:11119
  • 3. Denef J., Poles of $p$-adic complex powers and Newton polyhedra, Nieuw Archief voor Wiskunde 13 (1995), 289-295. MR 96m:11106
  • 4. Denef J., Hoornaert Kathleen, Newton polyhedra and Igusa local zeta function, J. Number Theory 89 (2001), no. 1, 31-64. MR 2002g:11170
  • 5. Igusa Jun-Ichi, An introduction to the theory of local zeta functions, AMS/IP studies in advanced mathematics, v. 14, 2000. MR 2001j:11112
  • 6. Igusa Jung-Ichi, A stationary phase formula for $p$-adic integrals and its applications, in Algebraic Geometry and its Applications, Springer-Verlag, New York, 1994, pp. 175-194. MR 95a:11104
  • 7. Igusa Jung-Ichi, Complex powers of irreducible algebroid curves, Geometry Today, Birkhäuser, Boston, 1985, pp. 207-230. MR 88j:11084
  • 8. Kempf G., Knudsen F., Mumford D., and Saint-Donat B., Toroidal Embeddings, Lecture Notes in Mathematics, vol. 339, Springer-Verlag, Berlin, 1973. MR 49:299
  • 9. Kouchnirenko A. G., Polyèdres de Newton et nombres de Milnor, Invent. Math. 32 (1976), 1-31. MR 54:7454
  • 10. Lichtin B., Meuser D., Poles of a local zeta function and Newton polygons, Compos. Math. 55 (1985), 313-332. MR 87a:11120
  • 11. Meuser D., On the poles of a local zeta function for curves, Invent. Math. 73 (1983), 445-465. MR 85i:14014
  • 12. Strauss L., Poles of two-variable $p$-adic complex power, Trans. Amer. Math. Soc. 27 (1983), 481-493. MR 84k:14019
  • 13. Varchenko A., Newton polyhedra and estimation of oscillanting integrals, Funct. Anal. Appl. 10 (1976), 175-196.
  • 14. Veys W., On the poles of Igusa's local zeta functions for curves, J. Lond. Math. Soc. 41 (1990), 27-32. MR 92j:11142
  • 15. Veys W., Poles of Igusa's local zeta function and monodromy, Bull. Soc. Math. Fr. 121 (1993), 545-598. MR 95b:11110
  • 16. Zuniga-Galindo W. A., Igusa's local zeta functions of semiquasihomogeneous polynomials, Trans. Amer. Math. Soc. 353 (2001), 3193-3207. MR 2001j:11116
  • 17. Zuniga-Galindo W. A., Local zeta functions and Newton polyhedra, to appear in Nagoya Math. J.
  • 18. Zuniga-Galindo W. A., Local zeta function for polynomial non-degenerate homogeneous mappings, preprint 2003.

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Additional Information

M. J. Saia
Affiliation: Instituto de Matemática E Computaçao, Universidade de São Paulo at São Carlos, Av. do Trabalhador São-Carlense 400, CEP 13560-970, São Carlos - SP, Brasil

W. A. Zuniga-Galindo
Affiliation: Department of Mathematics and Computer Science, Barry University, 11300 N.E. Second Avenue, Miami Shores, Florida 33161

Keywords: Igusa local zeta functions, Newton polygons, degenerate curves, non-degeneracy conditions, polynomial congruences
Received by editor(s): July 10, 2001
Received by editor(s) in revised form: May 6, 2003
Published electronically: December 15, 2003
Additional Notes: The first named author was partially supported by CNPq-Grant 300556/92-6
The second named author was supported by COLCIENCIAS-Grant # 089-2000. The second named author also thanks the partial support given by FAPESP for visiting the Instituto de Matemática e Computaçao, Universidade de São Paulo, Campus São Carlos, in January 2000
Article copyright: © Copyright 2003 American Mathematical Society

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