Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-29T06:08:44.107Z Has data issue: false hasContentIssue false
  • English
  • Français

Computing zeta functions of nondegenerate hypersurfaces with few monomials

Part of: Families, fibrations (Co)homology theory Zeta and $L$-functions: analytic theory Computational number theory

Published online by Cambridge University Press:  14 February 2013

Steven Sperber  and
John Voight
Steven Sperber
Affiliation:
School of Mathematics, University of Minnesota, 127 Vincent Hall Minneapolis 55455, USA (email: [email protected])
John Voight
Affiliation:
Department of Mathematics and Statistics, University of Vermont, 16 Colchester Avenue Burlington VT 05401-1455, USA (email: [email protected])

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Using the cohomology theory of Dwork, as developed by Adolphson and Sperber, we exhibit a deterministic algorithm to compute the zeta function of a nondegenerate hypersurface defined over a finite field. This algorithm is particularly well suited to work with polynomials in small characteristic that have few monomials (relative to their dimension). Our method covers toric, affine, and projective hypersurfaces, and also can be used to compute the L-function of an exponential sum.

MSC classification

Secondary: 11Y16: Algorithms; complexity 11M38: Zeta and $L$-functions in characteristic $p$ 14D10: Arithmetic ground fields (finite, local, global) 14F30: $p$-adic cohomology, crystalline cohomology
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 16 , October 2013 , pp. 9 - 44
Copyright
© The Author(s) 2013

References

[1] Abbott, T., Kedlaya, K. and Roe, D., ‘Bounding Picard numbers of surfaces using p-adic cohomology’, Arithmetic, geometry and coding theory (AGCT 2005), Séminaires et Congrès 21 (Société Mathématique de France, 2009) 125–159.Google Scholar
[2] Adolphson, A. and Sperber, S., ‘Exponential sums and Newton polyhedra: cohomology and estimates’, Ann. of Math. (2) 130 (1989) 367–406.CrossRefGoogle Scholar
[3] Adolphson, A. and Sperber, S., ‘p-adic estimates for exponential sums’, p-adic analysis (Trento, 1989), Lecture Notes in Mathematics 1454 (Springer, Berlin, 1990) 11–22.CrossRefGoogle Scholar
[4] Adolphson, A. and Sperber, S., ‘Exponential sums nondegenerate relative to a lattice’, Algebra Number Theory 3 (2009) 881–906.CrossRefGoogle Scholar
[5] Batyrev, V. and Cox, D., ‘On the Hodge structure of projective hypersurfaces in toric varieties’, Duke Math. J. 75 (1994) 293–338.CrossRefGoogle Scholar
[6] Bates (IMA), D. J., Bihan, F. and Sottile, F., ‘Bounds on the number of real solutions to polynomial equations’, Preprint, 2007, arXiv:math.AG/0706.4134.Google Scholar
[7] Bostan, A., Gaudry, P. and Schost, E., ‘Linear recurrences with polynomial coefficients and application to integer factorization and Cartier–Manin operator’, SIAM J. Comput. 36 (2007) 1777–1806.CrossRefGoogle Scholar
[8] Bruns, W. and Koch, R., ‘Computing the integral closure of an affine semigroup’, Univ. Iagel. Acta Math. 39 (2001) 59–70.Google Scholar
[9] Castryck, W., ‘Point counting on nondegenerate curves’, PhD Thesis, Katholieke Universiteit, Leuven, 2006.Google Scholar
[10] Castryck, W., Denef, J. and Vercauteren, F., ‘Computing zeta functions of nondegenerate curves’, Int. Math. Res. Pap. IMRP 2006 (2006) 72017.Google Scholar
[11] Castryck, W. and Voight, J., ‘On nondegeneracy of curves’, Algebra Number Theory 3 (2009) 255–281.CrossRefGoogle Scholar
[12] Chazelle, B., ‘An optimal convex hull algorithm in any fixed dimension’, Discrete Comput. Geom. 10 (1993) 377–409.CrossRefGoogle Scholar
[13] Delsarte, J., ‘Nombre de solutions des équations polynomiales sur un corps fini’, Séminaire Bourbaki, vol. 1, Exp. No. 39 (1948–1951) (Société Mathématique de France, Paris, 1995) 321–329.Google Scholar
[14] Denef, J. and Loeser, F., ‘Weights of exponential sums, intersection cohomology, and Newton polyhedra’, Invent. Math. 106 (1991) 275–294.CrossRefGoogle Scholar
[15] Domich, P. D., Kannan, R. and Trotter, L. E. Jr., ‘Hermite Normal Form computation using modulo determinant arithmetic’, Math. Oper. Res. 12 (1987) 50–59.CrossRefGoogle Scholar
[16] Dwork, B., ‘On the rationality of the zeta function of an algebraic variety’, Amer. J. Math. 82 (1960) 631–648.CrossRefGoogle Scholar
[17] Dwork, B., ‘On the zeta function of a hypersurface’, Publ. Math. Inst. Hautes Études Sci. 12 (1962) 5–68.CrossRefGoogle Scholar
[18] Dwork, B., ‘A deformation theory for the zeta function of a hypersurface’, Proceedings of the International Congress of Mathematicians, Stockholm, 1962 (Institut Mittag-Leffler, Djursholm, Sweden, 1963) 247–259.Google Scholar
[19] Dwork, B., ‘On p-adic analysis’, Some recent advances in the basic sciences, vol. 2, Proc. Annual Sci. Conf., New York, 1965–1966 (Belfer Graduate School of Science, Yeshiva University, New York, 1969) 129–154.Google Scholar
[20] Dwork, B., ‘ p-adic cycles’, Publ. Math. Inst. Hautes Études Sci. 37 (1969) 27–115.CrossRefGoogle Scholar
[21] Edixhoven, B., Point counting after Kedlaya, lecture notes, EIDMA-Stieltjes graduate course, Leiden, 2003.Google Scholar
[22] Edixhoven, B., Couveignes, J.-M., de Jong, R., Merkl, F. and Bosman, J., ‘Computational aspects of modular forms and Galois representations’, Preprint, 2010, arXiv:math.NT/0605244.CrossRefGoogle Scholar
[23] Fortune, S., ‘Voronoi diagrams and Delaunay triangulations’, Computing in Euclidean geomtry (eds Du, D.-Z. and Hwang, F. K.; World Scientific, 1995) 193–233.Google Scholar
[24] Fortune, S., ‘Voronoi diagrams and Delaunay triangulations’, Handbook of discrete and computational geometry (eds Goodman, J. E. and O’Rourke, J.; CRC Press, Boca Raton, 1997) 377–388.Google Scholar
[25] Gerkmann, R., ’Relative rigid cohomology and deformation of hypersurfaces’, Int. Math. Res. Pap. IMRP 2007 (2007) rpm003.Google Scholar
[26] Gritzmann, P., Klee, V. and Larman, D. G., ‘Largest j-simplices n-polytopes’, Discrete Comput. Geom. 13 (1995) 477–515.CrossRefGoogle Scholar
[27] Harvey, D., ‘Kedlaya’s algorithm in larger characteristic’, Int. Math. Res. Not. IMRN 2007 (2007) rnm095.Google Scholar
[28] Harvey, D., ‘Computing zeta functions of projective surfaces in large characteristic’, Preprint.Google Scholar
[29] Howell, R. R., ‘On asymptotic notation with multiple variables’, Technical Report 2007-4, Preprint, 2008, http://people.cis.ksu.edu/∼rhowell/asymptotic.pdf.Google Scholar
[30] Kadir, S., ‘The arithmetic of Calabi–Yau manifolds and mirror symmetry’, PhD Thesis, University of Oxford, Oxford, 2004.Google Scholar
[31] Kannan, R. and Bachem, A., ‘Polynomial algorithms for computing the Smith and Hermite normal forms of an integer matrix’, SIAM J. Comput. 8 (1979) 499–507.CrossRefGoogle Scholar
[32] Katz, N. M., ‘On the differential equations satisfied by period matrices’, Publ. Math. Inst. Hautes Études Sci. 35 (1968) 71–106.CrossRefGoogle Scholar
[33] Katz, N. M. and Sarnak, P., Random matrices, Frobenius eigenvalues, and monodromy (American Mathematical Society, Providence, RI, 1998).CrossRefGoogle Scholar
[34] Kedlaya, K., ‘Counting points on hyperelliptic curves using Monsky–Washnitzer cohomology’, J. Ramanujan Math. Soc. 16 (2001) 323–338.Google Scholar
[35] Kedlaya, K., ‘Search techniques for root-unitary polynomials’, Computational arithmetic geometry, Contemporary Mathematics 463 (American Mathematical Society, Providence, RI, 2008) 71–81.CrossRefGoogle Scholar
[36] Kedlaya, K., ‘Computing zeta functions of nondegenerate toric hypersurfaces: two proposals’, Preprint.Google Scholar
[37] Khovanskii, A. G., ‘Newton polyhedra and toroidal varieties’, Funct. Anal. Appl. 11 (1977) 289–296.CrossRefGoogle Scholar
[38] Koblitz, N., p-adic numbers, p-adic analysis, and zeta-functions, Graduate Texts in Mathematics 58 (Springer, 1977).CrossRefGoogle Scholar
[39] Koblitz, N., ‘The number of points on certain families of hypersurfaces over finite fields’, Compositio Math. 48 (1983) 3–23.Google Scholar
[40] Kouchnirenko, A. G., ‘Polyèdres de Newton et nombres de Milnor’, Inv. Math. 32 (1976) 1–31.CrossRefGoogle Scholar
[41] Kouchnirenko, A. G., Fewnomials, Translations of Mathematical Monographs 88 (American Mathematical Society, Providence, RI, 1991).Google Scholar
[42] Lagarias, J. and Ziegler, G., ‘Bound for lattice polytopes containing a fixed number of interior points in a sublattice’, Canad. J. Math. 43 (1991) 1022–1035.CrossRefGoogle Scholar
[43] Lauder, A. G. B., ‘Deformation theory and the computation of zeta functions’, Proc. Lond. Math. Soc. (3) 88 (2004) 565–602.CrossRefGoogle Scholar
[44] Lauder, A. G. B., ‘Counting solutions to equations in many variables over finite fields’, Found. Comput. Math. 4 (2004) 221–267.CrossRefGoogle Scholar
[45] Lauder, A. G. B., ‘A recursive method for computing zeta functions of varieties’, LMS J. Comput. Math. 9 (2006) 222–269.CrossRefGoogle Scholar
[46] Lauder, A. G. B. and Wan, D., ‘Computing zeta functions of Artin–Schreier curves over finite fields’, LMS J. Comput. Math. 5 (2002) 34–55.CrossRefGoogle Scholar
[47] Lauder, A. G. B. and Wan, D., ‘Computing zeta functions of Artin–Schreier curves over finite fields. II’, J. Complexity 20 (2004) 331–349.CrossRefGoogle Scholar
[48] Lauder, A. G. B. and Wan, D., ‘Counting points on varieties over finite fields of small characteristic’, Algorithmic number theory: lattices, number fields, curves and cryptography, Mathematical Sciences Research Institute Publications 44 (Cambridge University Press, Cambridge, 2008) 579–612.Google Scholar
[49] de Loera, J., Hemmecke, R. and Köppe, M., ‘Pareto optima of multicriteria integer programs’, INFORMS J. Comput. 21 (2009) 39–48.CrossRefGoogle Scholar
[50] Mavlyutov, A. R., ‘Cohomology of complete intersections in toric varieties’, Pacific J. Math. 191 (1999) 133–144.CrossRefGoogle Scholar
[51] Monsky, P., p-adic analysis and zeta functions, Lectures in Mathematics (Kyoto University, Tokyo, 1970) http://www.math.kyoto-u.ac.jp/library/lmku/lmku04.pdf.Google Scholar
[52] Preparata, F. P. and Shamos, M. I., Computational geometry: an introduction, 3rd edn (Springer, 1991).Google Scholar
[53] Savitt, D., Thakur, D., Baker, M., Conrad, B., Dasgupta, S., Kedlaya, K. and Teitelbaum, J., p-adic geometry: lectures from the 2007 Arizona winter school, University Lecture Series 45 (American Mathematical Society, Providence, RI, 2008).Google Scholar
[54] Schoof, R., ‘Elliptic curves over finite fields and the computation of square roots mod p ’, Math. Comp. 44 (1985) 483–494.Google Scholar
[55] Seidel, R., ‘Convex hull computations’, Handbook of discrete and computational geometry (eds Goodman, J. E. and O’Rourke, J.; CRC Press, Boca Raton, FL, 1997) 361–375.Google Scholar
[56] Serre, J.-P., ‘Zeta and L-functions’, Arithmetic algebraic geometry (ed. Schilling, ; Harper and Row, New York, NY, 1965).Google Scholar
[57] Tuitman, J., ‘Counting points in families of nondegenerate curves’, PhD Thesis, Katholieke Universiteit, Leuven, 2010.Google Scholar
[58] Wan, D., ‘Computing zeta functions over finite fields’, Contemp. Math. 225 (1999) 131–141.CrossRefGoogle Scholar
[59] Wan, D., ‘Modular counting of rational points over finite fields’, Found. Comput. Math. 8 (2008) 597–605.CrossRefGoogle Scholar
[60] Wan, D., ‘Algorithmic theory of zeta functions over finite fields’, Algorithmic number theory: lattices, number fields, curves and cryptography, Mathematical Sciences Research Institute Publications 44 (Cambridge University Press, Cambridge, 2008) 551–578.Google Scholar
[61] Weil, A., ‘Numbers of solutions of equations in finite fields’, Bull. Amer. Math. Soc. 55 (1949) 497–508.CrossRefGoogle Scholar
[62] Widmer, M., ‘Lipschitz class, narrow class, and counting lattice points’, Proc. Amer. Math. Soc., to appear.Google Scholar
[63] Wong, C. F., ‘Zeta functions of projective toric hypersurfaces over finite fields’, PhD thesis, University of California, Irvine, 2008.Google Scholar