Local zeta function for curves, non-degeneracy conditions and Newton polygons
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- by M. J. Saia and W. A. Zuniga-Galindo PDF
- Trans. Amer. Math. Soc. 357 (2005), 59-88 Request permission
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 $Z(s,f,v)$.References
- Arnold V., Varchenko A. and Gussein-Zade S., Singularités des applications différentiables, vol. 2, Mir, Moscow, 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, Poles of $p$-adic complex powers and Newton polyhedra, Nieuw Arch. Wisk. (4) 13 (1995), no. 3, 289–295. MR 1378800
- 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
- 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
- Jun-ichi Igusa, A stationary phase formula for $p$-adic integrals and its applications, Algebraic geometry and its applications (West Lafayette, IN, 1990) Springer, New York, 1994, pp. 175–194. MR 1272029
- 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
- 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
- A. G. Kouchnirenko, Polyèdres de Newton et nombres de Milnor, Invent. Math. 32 (1976), no. 1, 1–31 (French). MR 419433, DOI 10.1007/BF01389769
- Ben Lichtin and Diane Meuser, Poles of a local zeta function and Newton polygons, Compositio Math. 55 (1985), no. 3, 313–332. MR 799820
- 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
- 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
- Varchenko A., Newton polyhedra and estimation of oscillanting integrals, Funct. Anal. Appl. 10 (1976), 175-196.
- 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
- W. A. Zúñiga-Galindo, Igusa’s local zeta functions of semiquasihomogeneous polynomials, Trans. Amer. Math. Soc. 353 (2001), no. 8, 3193–3207. MR 1608309, DOI 10.1090/S0002-9947-01-02323-6
- Zuniga-Galindo W. A., Local zeta functions and Newton polyhedra, to appear in Nagoya Math. J.
- Zuniga-Galindo W. A., Local zeta function for polynomial non-degenerate homogeneous mappings, preprint 2003.
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
- MR Author ID: 308611
- Email: mjsaia@icmc.usp.br
- W. A. Zuniga-Galindo
- Affiliation: Department of Mathematics and Computer Science, Barry University, 11300 N.E. Second Avenue, Miami Shores, Florida 33161
- Email: wzuniga@mail.barry.edu
- 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 - © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 357 (2005), 59-88
- MSC (2000): Primary 11D79, 14G20, 14M25
- DOI: https://doi.org/10.1090/S0002-9947-03-03491-3
- MathSciNet review: 2098087