Newton polyhedra, unstable faces and the poles of Igusa’s local zeta function

Author:
Kathleen Hoornaert

Journal:
Trans. Amer. Math. Soc. **356** (2004), 1751-1779

MSC (2000):
Primary 11S40, 11D79; Secondary 14M25, 52B20, 14G10

DOI:
https://doi.org/10.1090/S0002-9947-03-03507-4

Published electronically:
December 15, 2003

MathSciNet review:
2031040

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we examine when the order of a pole of Igusa’s local zeta function associated to a polynomial $f$ is smaller than “expected”. We carry out this study in the case that $f$ is sufficiently non-degenerate with respect to its Newton polyhedron $\Gamma (f)$, and the main result of this paper is a proof of one of the conjectures of Denef and Sargos. Our technique consists in reducing our question about the polynomial $f$ to the same question about polynomials $f_\mu$, where $\mu$ are faces of $\Gamma (f)$ depending on the examined pole and $f_\mu$ is obtained from $f$ by throwing away all monomials of $f$ whose exponents do not belong to $\mu$. Secondly, we obtain a formula for Igusa’s local zeta function associated to a polynomial $f_\mu$, with $\mu$ unstable, which shows that, in this case, the upperbound for the order of the examined pole is obviously smaller than “expected”.

- 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 https://doi.org/10.1007/BF01389133 - J. Denef,
*On the degree of Igusa’s local zeta function*, Amer. J. Math.**109**(1987), no. 6, 991–1008. MR**919001**, DOI https://doi.org/10.2307/2374583 - 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 https://doi.org/10.1006/jnth.2000.2606 - J. Denef, A. Laeremans, and P. Sargos,
*On the largest nontrivial pole of the distribution $|f|^s$*, Sūrikaisekikenkyūsho K\B{o}kyūroku**999**(1997), 1–9. Research on prehomogeneous vector spaces (Japanese) (Kyoto, 1996). MR**1622331** - Jan Denef and Patrick Sargos,
*Polyèdre de Newton et distribution $f^s_+$. II*, Math. Ann.**293**(1992), no. 2, 193–211 (French). MR**1166118**, DOI https://doi.org/10.1007/BF01444712 - K. Hoornaert and D. Loots,
*Polygusa: a computer program for Igusa’s local zeta function*, http://www.wis.kuleuven.ac.be/wis/algebra/kathleen.htm, 2000. - K. Hoornaert,
*Newton polyhedra and the poles of Igusa’s local zeta function*, Bull. Belg. Math. Soc. - Simon Stevin**9**(2002), 589–606. - J.-I. Igusa,
*Complex powers and asympotic expansions I*, J. Reine Angew. Math**268/269**(1974), 110–130; II, ibid, 278/279, 307–321,1975. ; - Jun-ichi Igusa,
*Forms of higher degree*, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 59, Tata Institute of Fundamental Research, Bombay; by the Narosa Publishing House, New Delhi, 1978. MR**546292** - 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. H. Phong and Jacob Sturm,
*Algebraic estimates, stability of local zeta functions, and uniform estimates for distribution functions*, Ann. of Math. (2)**152**(2000), no. 1, 277–329. MR**1792297**, DOI https://doi.org/10.2307/2661384 - D. H. Phong, E. M. Stein, and J. A. Sturm,
*On the growth and stability of real-analytic functions*, Amer. J. Math.**121**(1999), no. 3, 519–554. MR**1738409** - R. Tyrrell Rockafellar,
*Convex analysis*, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR**0274683** - W.A. Zúñiga-Galindo,
*Local zeta functions and Newton polyhedra*, preprint, 1999.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
11S40,
11D79,
14M25,
52B20,
14G10

Retrieve articles in all journals with MSC (2000): 11S40, 11D79, 14M25, 52B20, 14G10

Additional Information

**Kathleen Hoornaert**

Affiliation:
Department of Mathematics, Catholic University Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium

Keywords:
Igusa zeta function,
Newton polyhedron,
congruences,
$p$-adic integrals

Received by editor(s):
March 12, 2002

Published electronically:
December 15, 2003

Article copyright:
© Copyright 2003
American Mathematical Society