The motivic Igusa zeta series of some hypersurfaces non-degenerated with respect to their Newton polyhedron
Abstract
We describe some algorithms, without using resolution of singularities, that establish the rationality of the motivic Igusa zeta series of certain hypersurfaces that are non-degenerated with respect to their Newton polyhedron. This includes, in any characteristic, the motivic rationality for polydiagonal hypersurfaces, vertex singularities, binomial hypersurfaces, and Du Val singularities.
Keywords
motivic Igusa zeta series; Du Val singularities; stationary phase method
References
2. Bosch S., Lütkebohmert W., Raynaud M., Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21, Springer-Verlag, Berlin, 1990.
3. Denef J., Report on Igusa’s local zeta function, Séminaire Bourbaki Vol. 1990/91, Astérisque (1991), no. 201–203, Exp. No. 741 (1992), 359–386.
4. Denef J., Hoornaert K., Newton polyhedra and Igusa’s local zeta function, J. Number Theory 89 (2001), no. 1, 31–64.
5. Denef J., Loeser F., Motivic Igusa zeta functions, J. Algebraic Geom. 7 (1998), no. 3, 505–537.
6. Denef J., Hoornaert K., Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math. 135 (1999), no. 1, 201–232.
7. Denef J., Hoornaert K., On some rational generating series occurring in arithmetic geometry, in: Geometric aspects of Dwork theory. Vol. I, II, Walter de Gruyter GmbH & Co. KG, Berlin, 2004, pp. 509–526.
8. Eisenbud D., Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995.
9. Guibert G., Espaces d’arcs et invariants d’Alexander, Comment. Math. Helv. 77 (2002), no. 4, 783–820.
10. Hartshorne R., Algebraic geometry, Springer-Verlag, New York, 1977.
11. Igusa J., A stationary phase formula for p-adic integrals and its applications, in: Algebraic geometry and its applications (West Lafayette, IN, 1990), Springer, New York, 1994, pp. 175–194.
12. Igusa J., An introduction to the theory of local zeta functions, AMS/IP Studies in Advanced Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2000.
13. Lichtin B., Meuser D., Poles of a local zeta function and Newton polygons, Compositio Math. 55 (1985), no. 3, 313–332.
14. Matsumura H., Commutative ring theory, Cambridge University Press, Cambridge, 1986.
15. Saia M.J., Zuniga-Galindo W.A., Local zeta function for curves, non-degeneracy conditions and Newton polygons, Trans. Amer. Math. Soc. 357 (2005), no. 1, 59–88.
16. Schoutens H., Classifying singularities up to analytic extensions of scalars is smooth, Ann. Pure Appl. Logic 162 (2011), 836–852.
17. Schoutens H., Schemic Grothendieck rings I: motivic sites, Preprint 2011.
18. Schoutens H., Schemic Grothendieck rings II: jet schemes and motivic integration, Preprint 2011.
19. Varchenko A., Zeta-function of monodromy and Newton’s diagram, Invent. Math. 37 (1976), no. 3, 253–262.
20. Veys W., Poles of Igusa’s local zeta function and monodromy, Bull. Soc. Math. France 121 (1993), no. 4, 545–598.
21. Veys W., Determination of the poles of the topological zeta function for curves, Manuscripta Math. 87 (1995), no. 4, 435–448.
22. Zúñiga-Galindo W., Igusa’s local zeta functions of semiquasihomogeneous polynomials, Trans. Amer. Math. Soc. 353 (2001), no. 8, 3193–3207.
23. Zúñiga-Galindo W., Local zeta functions and Newton polyhedra, Nagoya Math. J. 172 (2003), 31–58.
Department of Mathematics, City University of New York, USA United States
![Creative Commons License](http://i.creativecommons.org/l/by/4.0/88x31.png)
This work is licensed under a Creative Commons Attribution 4.0 International License.
The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.
- License
This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license. - Author’s Warranties
The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s. - User Rights
Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor. - Co-Authorship
If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.