On the poles of topological zeta functions
HTML articles powered by AMS MathViewer
- by Ann Lemahieu, Dirk Segers and Willem Veys PDF
- Proc. Amer. Math. Soc. 134 (2006), 3429-3436 Request permission
Abstract:
We study the topological zeta function $Z_{top,f}(s)$ associated to a polynomial $f$ with complex coefficients. This is a rational function in one variable, and we want to determine the numbers that can occur as a pole of some topological zeta function; by definition these poles are negative rational numbers. We deal with this question in any dimension. Denote $\mathcal {P}_n := \{ s_0 \mid \exists f \in \mathbb {C}[x_1,\ldots , x_n] : Z_{top,f}(s)$ has a pole in $s_0 \}$. We show that $\{-(n-1)/2-1/i \mid i \in \mathbb {Z}_{>1}\}$ is a subset of $\mathcal {P}_n$; for $n=2$ and $n=3$, the last two authors proved before that these are exactly the poles less than $-(n-1)/2$. As the main result we prove that each rational number in the interval $[-(n-1)/2,0)$ is contained in $\mathcal {P}_n$.References
- Valery Alexeev, Boundedness and $K^2$ for log surfaces, Internat. J. Math. 5 (1994), no. 6, 779–810. MR 1298994, DOI 10.1142/S0129167X94000395
- J. Denef and F. Loeser, Caractéristiques d’Euler-Poincaré, fonctions zêta locales et modifications analytiques, J. Amer. Math. Soc. 5 (1992), no. 4, 705–720 (French). MR 1151541, DOI 10.1090/S0894-0347-1992-1151541-7
- Jan Denef and François Loeser, Motivic Igusa zeta functions, J. Algebraic Geom. 7 (1998), no. 3, 505–537. MR 1618144
- Jan Denef and François Loeser, Motivic exponential integrals and a motivic Thom-Sebastiani theorem, Duke Math. J. 99 (1999), no. 2, 285–309. MR 1708026, DOI 10.1215/S0012-7094-99-09910-6
- 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
- K. Hoornaert and D. Loots, A computer program written in Maple to calculate Igusa’s $p$-adic zeta function and the topological zeta funtion for non-degenerated polynomials, available on http://www.wis.kuleuven.be/algebra/kathleen.htm (2002).
- János Kollár, Log surfaces of general type; some conjectures, Classification of algebraic varieties (L’Aquila, 1992) Contemp. Math., vol. 162, Amer. Math. Soc., Providence, RI, 1994, pp. 261–275. MR 1272703, DOI 10.1090/conm/162/01538
- János Kollár, Singularities of pairs, Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 221–287. MR 1492525
- Takayasu Kuwata, On log canonical thresholds of reducible plane curves, Amer. J. Math. 121 (1999), no. 4, 701–721. MR 1704476, DOI 10.1353/ajm.1999.0028
- Takayasu Kuwata, On log canonical thresholds of surfaces in $\textbf {C}^3$, Tokyo J. Math. 22 (1999), no. 1, 245–251. MR 1692033, DOI 10.3836/tjm/1270041625
- James McKernan and Yuri Prokhorov, Threefold thresholds, Manuscripta Math. 114 (2004), no. 3, 281–304. MR 2075967, DOI 10.1007/s00229-004-0457-x
- Yu. G. Prokhorov, On log canonical thresholds, Comm. Algebra 29 (2001), no. 9, 3961–3970. Special issue dedicated to Alexei Ivanovich Kostrikin. MR 1857023, DOI 10.1081/AGB-100105984
- Yu. G. Prokhorov, On log canonical thresholds. II, Comm. Algebra 30 (2002), no. 12, 5809–5823. MR 1941925, DOI 10.1081/AGB-120016015
- D. Segers, Smallest poles of Igusa’s and topological zeta functions and solutions of polynomial congruences, K.U. Leuven Ph.D. thesis, available on http://www.wis.kuleuven.be/algebra/segers/segers.htm (2004).
- D. Segers, Lower bound for the poles of Igusa’s p-adic zeta functions, Math. Annalen (to appear).
- V. V. Shokurov, Three-dimensional log perestroikas, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 1, 105–203 (Russian); English transl., Russian Acad. Sci. Izv. Math. 40 (1993), no. 1, 95–202. MR 1162635, DOI 10.1070/IM1993v040n01ABEH001862
- Dirk Segers and Willem Veys, On the smallest poles and topological zeta functions, Compos. Math. 140 (2004), no. 1, 130–144. MR 2004126, DOI 10.1112/S0010437X03000344
- 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, DOI 10.24033/bsmf.2219
- Willem Veys, Determination of the poles of the topological zeta function for curves, Manuscripta Math. 87 (1995), no. 4, 435–448. MR 1344599, DOI 10.1007/BF02570485
Additional Information
- Ann Lemahieu
- Affiliation: Departement Wiskunde, Celestijnenlaan 200B, B-3001 Leuven, Belgium
- Email: ann.lemahieu@wis.kuleuven.be
- Dirk Segers
- Affiliation: Departement Wiskunde, Celestijnenlaan 200B, B-3001 Leuven, Belgium
- Email: dirk.segers@wis.kuleuven.be
- Willem Veys
- Affiliation: Departement Wiskunde, Celestijnenlaan 200B, B-3001 Leuven, Belgium
- Email: wim.veys@wis.kuleuven.be
- Received by editor(s): January 25, 2005
- Received by editor(s) in revised form: June 29, 2005
- Published electronically: June 9, 2006
- Communicated by: Michael Stillman
- © Copyright 2006 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 134 (2006), 3429-3436
- MSC (2000): Primary 14B05, 14J17, 32S05; Secondary 14E15, 32S25
- DOI: https://doi.org/10.1090/S0002-9939-06-08512-1
- MathSciNet review: 2240652