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Counting points on varieties over finite fields of small characteristic
 ALGORITHMIC NUMBER THEORY
, 2008
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Modular Counting of Rational Points over Finite Fields
"... Let Fq be the finite field of q elements, where q = p h. Let f(x) be a polynomial over Fq in n variables with m nonzero terms. Let N(f) denote the number of solutions of f(x) = 0 with coordinates in Fq. In this paper, we give a deterministic algorithm which computes the reduction of N(f) modulo p ..."
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Let Fq be the finite field of q elements, where q = p h. Let f(x) be a polynomial over Fq in n variables with m nonzero terms. Let N(f) denote the number of solutions of f(x) = 0 with coordinates in Fq. In this paper, we give a deterministic algorithm which computes the reduction of N(f) modulo p b in O(n(8m) (h+b)p) bit operations. This is singly exponential in each of the parameters {h, b, p}, answering affirmatively an open problem proposed in [5]. 1
Computing zeta functions of nondegenerate hypersurfaces with few monomials
 SUBMITTED EXCLUSIVELY TO THE LONDON MATHEMATICAL SOCIETY
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Geometric moment zeta functions
 In Geometric Aspects of Dwork Theory, Walter de Gruyter
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COMPUTING ZETA FUNCTIONS OF SPARSE NONDEGENERATE HYPERSURFACES
"... 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 wellsuited to work with polynomials in small character ..."
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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 wellsuited 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 Lfunction of an exponential sum. Let p be prime and let Fq be a finite field with q = p a elements. Let V be a variety defined over Fq, described by the vanishing of a finite set of polynomial equations with coefficients in Fq. We encode the number of points #V (Fqr) on V over the extensions Fqr of Fq in an exponential generating series, called the zeta function of V:
Computing Igusa's Local Zeta Functions of Univariate Polynomials, and Linear Feedback Shift Registers
, 2003
"... We give a polynomial time algorithm for computing the Igusa local zeta function Z(s; f) attached to a polynomial f(x) 2 Z[x], in one variable, with splitting eld Q, and a prime number p. We also propose a new class of linear feedback shift registers based on the computation of Igusa's local ze ..."
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We give a polynomial time algorithm for computing the Igusa local zeta function Z(s; f) attached to a polynomial f(x) 2 Z[x], in one variable, with splitting eld Q, and a prime number p. We also propose a new class of linear feedback shift registers based on the computation of Igusa's local zeta function.
CalabiYau hypersurfaces
, 2004
"... The aim of this course is to apply certain recent developments in Dwork’s padic theory to study the padic variation of the zeta function attached to a family of affine toric CalabiYau hypersurfaces over finite fields, leading up to Dwork’s unit root conjecture. ..."
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The aim of this course is to apply certain recent developments in Dwork’s padic theory to study the padic variation of the zeta function attached to a family of affine toric CalabiYau hypersurfaces over finite fields, leading up to Dwork’s unit root conjecture.