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Computing zeta functions of nondegenerate hypersurfaces with few monomials

  • Steven Sperber (a1) and John Voight (a2)
Abstract
Abstract

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.

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[2] A. Adolphson and S. Sperber , ‘Exponential sums and Newton polyhedra: cohomology and estimates’, Ann. of Math. (2) 130 (1989) 367406.

[3] A. Adolphson and S. Sperber , ‘p-adic estimates for exponential sums’, p-adic analysis (Trento, 1989), Lecture Notes in Mathematics 1454 (Springer, Berlin, 1990) 1122.

[4] A. Adolphson and S. Sperber , ‘Exponential sums nondegenerate relative to a lattice’, Algebra Number Theory 3 (2009) 881906.

[5] V. Batyrev and D. Cox , ‘On the Hodge structure of projective hypersurfaces in toric varieties’, Duke Math. J. 75 (1994) 293338.

[7] A. Bostan , P. Gaudry and E. Schost , ‘Linear recurrences with polynomial coefficients and application to integer factorization and Cartier–Manin operator’, SIAM J. Comput. 36 (2007) 17771806.

[11] W. Castryck and J. Voight , ‘On nondegeneracy of curves’, Algebra Number Theory 3 (2009) 255281.

[12] B. Chazelle , ‘An optimal convex hull algorithm in any fixed dimension’, Discrete Comput. Geom. 10 (1993) 377409.

[14] J. Denef and F. Loeser , ‘Weights of exponential sums, intersection cohomology, and Newton polyhedra’, Invent. Math. 106 (1991) 275294.

[15] P. D. Domich , R. Kannan and L. E. Trotter Jr., ‘Hermite Normal Form computation using modulo determinant arithmetic’, Math. Oper. Res. 12 (1987) 5059.

[16] B. Dwork , ‘On the rationality of the zeta function of an algebraic variety’, Amer. J. Math. 82 (1960) 631648.

[17] B. Dwork , ‘On the zeta function of a hypersurface’, Publ. Math. Inst. Hautes Études Sci. 12 (1962) 568.

[20] B. Dwork , ‘p-adic cycles’, Publ. Math. Inst. Hautes Études Sci. 37 (1969) 27115.

[26] P. Gritzmann , V. Klee and D. G. Larman , ‘Largest j-simplices n-polytopes’, Discrete Comput. Geom. 13 (1995) 477515.

[31] R. Kannan and A. Bachem , ‘Polynomial algorithms for computing the Smith and Hermite normal forms of an integer matrix’, SIAM J. Comput. 8 (1979) 499507.

[32] N. M. Katz , ‘On the differential equations satisfied by period matrices’, Publ. Math. Inst. Hautes Études Sci. 35 (1968) 71106.

[35] K. Kedlaya , ‘Search techniques for root-unitary polynomials’, Computational arithmetic geometry, Contemporary Mathematics 463 (American Mathematical Society, Providence, RI, 2008) 7181.

[37] A. G. Khovanskii , ‘Newton polyhedra and toroidal varieties’, Funct. Anal. Appl. 11 (1977) 289296.

[38] N. Koblitz , p-adic numbers, p-adic analysis, and zeta-functions, Graduate Texts in Mathematics 58 (Springer, 1977).

[40] A. G. Kouchnirenko , ‘Polyèdres de Newton et nombres de Milnor’, Inv. Math. 32 (1976) 131.

[42] J. Lagarias and G. Ziegler , ‘Bound for lattice polytopes containing a fixed number of interior points in a sublattice’, Canad. J. Math. 43 (1991) 10221035.

[43] A. G. B. Lauder , ‘Deformation theory and the computation of zeta functions’, Proc. Lond. Math. Soc. (3) 88 (2004) 565602.

[44] A. G. B. Lauder , ‘Counting solutions to equations in many variables over finite fields’, Found. Comput. Math. 4 (2004) 221267.

[45] A. G. B. Lauder , ‘A recursive method for computing zeta functions of varieties’, LMS J. Comput. Math. 9 (2006) 222269.

[46] A. G. B. Lauder and D. Wan , ‘Computing zeta functions of Artin–Schreier curves over finite fields’, LMS J. Comput. Math. 5 (2002) 3455.

[47] A. G. B. Lauder and D. Wan , ‘Computing zeta functions of Artin–Schreier curves over finite fields. II’, J. Complexity 20 (2004) 331349.

[49] J. de Loera , R. Hemmecke and M. Köppe , ‘Pareto optima of multicriteria integer programs’, INFORMS J. Comput. 21 (2009) 3948.

[50] A. R. Mavlyutov , ‘Cohomology of complete intersections in toric varieties’, Pacific J. Math. 191 (1999) 133144.

[54] R. Schoof , ‘Elliptic curves over finite fields and the computation of square roots mod p’, Math. Comp. 44 (1985) 483494.

[58] D. Wan , ‘Computing zeta functions over finite fields’, Contemp. Math. 225 (1999) 131141.

[59] D. Wan , ‘Modular counting of rational points over finite fields’, Found. Comput. Math. 8 (2008) 597605.

[61] A. Weil , ‘Numbers of solutions of equations in finite fields’, Bull. Amer. Math. Soc. 55 (1949) 497508.

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