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

Hoornaert, Kathleen

doi:10.1090/S0002-9947-03-03507-4

Newton polyhedra, unstable faces and the poles of Igusa’s local zeta function
HTML articles powered by AMS MathViewer

by Kathleen Hoornaert PDF

Trans. Amer. Math. Soc. 356 (2004), 1751-1779 Request permission

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”.

References

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 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 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 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 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.
Arnaud Denjoy, Sur certaines séries de Taylor admettant leur cercle de convergence comme coupure essentielle, C. R. Acad. Sci. Paris 209 (1939), 373–374 (French). MR 50
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; 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 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.