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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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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:
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.