Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On Igusa zeta functions of monomial ideals


Authors: Jason Howald, Mircea Mustata and Cornelia Yuen
Journal: Proc. Amer. Math. Soc. 135 (2007), 3425-3433
MSC (2000): Primary 14B05; Secondary 14M25
DOI: https://doi.org/10.1090/S0002-9939-07-08957-5
Published electronically: August 6, 2007
MathSciNet review: 2336554
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that the real parts of the poles of the Igusa zeta function of a monomial ideal can be computed from the torus-invariant divisors on the normalized blow-up of the affine space along the ideal. Moreover, we show that every such number is a root of the Bernstein-Sato polynomial associated to the monomial ideal.


References [Enhancements On Off] (What's this?)

  • [Be] J. N. Bernstein, Analytic continuation of generalized functions with respect to a parameter, Funk. Anal. 6 (1972), 26-40. MR 0320735 (47:9269)
  • [BMS1] N. Budur, M. Mustata and M. Saito, Combinatorial description of the roots of the Bernstein-Sato polynomials for monomial ideals, Comm. Algebra. 34 (2006), 4103-4117. MR 2267574 (2007h:32041)
  • [BMS2] N. Budur, M. Mustata and M. Saito, Roots of Bernstein-Sato polynomials for monomial ideals: a positive characteristic approach, Math. Res. Lett. 13 (2006), 125-142. MR 2200051 (2006k:14003)
  • [BMS3] N. Budur, M. Mustata and M. Saito, Bernstein-Sato polynomials of arbitrary varieties, Compos. Math. 142 (2006), 779-797. MR 2231202 (2007c:32036)
  • [De] J. Denef, Report on Igusa's local zeta function, Séminaire Bourbaki, Vol. 1990/91, Astérisque No. 201-203 (1991), Exp. No. 741, 359-386. MR 1157848 (93g:11119)
  • [DH] J. Denef and K. Hoornaert, Newton polyhedra and Igusa's local zeta function, J. Number Theory 89 (2001), 31-64. MR 1838703 (2002g:11170)
  • [DL] J. Denef and F. Loeser, Motivic Igusa zeta functions, J. Algebraic Geom. 7 (1998), 505-537. MR 1618144 (99j:14021)
  • [Fu] W. Fulton, Introduction to toric varieties, Annals of Mathematics Studies 131, The William H. Roever Lectures in Geometry, Princeton University Press, Princeton, NJ, 1993. MR 1234037 (94g:14028)
  • [Ig] J.-i. Igusa, An introduction to the theory of local zeta functions, AMS/IP Studies in Advanced Mathematics, 14, American Mathematical Society, Providence, RI; International Press, Cambridge, MA, 2000. MR 1743467 (2001j:11112)
  • [Lo] F. Loeser, Fonctions d'Igusa $ p$-adiques, polynômes de Bernstein, et polyhédres de Newton, J. Reine Angew. Math. 412 (1990), 75-96. MR 1079002 (92c:11139)
  • [SS] M. Sato and T. Shintani, On zeta functions associated with prehomogeneous vector spaces, Proc. Nat. Acad. Sci. U.S.A. 69 (1972), 1081-1082. MR 0296079 (45:5140)
  • [Ve] W. Veys, Embedded resolution of singularities and Igusa's local zeta function, Academiae Analecta (2001), available at wis.kuleuven.be/algebra/veys.htm.
  • [Zu] W. A. Zú $ {\rm\tilde{n}}$iga-Galindo, On the poles of the Igusa zeta function for algebraic sets, Bull. London Math. Soc. 36 (2004), 310-320. MR 2038719 (2005c:11149)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 14B05, 14M25

Retrieve articles in all journals with MSC (2000): 14B05, 14M25


Additional Information

Jason Howald
Affiliation: Department of Mathematics and Computer Science, John Carroll University, 20700 North Park Blvd., University Heights, Ohio 44118
Address at time of publication: Department of Mathematics, SUNY Potsdam, 44 Pierrepont Avenue, Potsdam, New York 13676-2294
Email: howaldja@potsdam.edu

Mircea Mustata
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: mmustata@umich.edu

Cornelia Yuen
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Address at time of publication: Department of Mathematics, University of Kentucky, 825 Patterson Office Tower, Lexington, Kentucky 40506
Email: cyuen@ms.uky.edu

DOI: https://doi.org/10.1090/S0002-9939-07-08957-5
Keywords: Igusa zeta function, monomial ideal, Bernstein-Sato polynomial
Received by editor(s): June 15, 2006
Published electronically: August 6, 2007
Additional Notes: The research of the second author was partially supported by NSF grant DMS 0500127
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society