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Computing zeta functions of nondegenerate hypersurfaces with few monomials
 SUBMITTED EXCLUSIVELY TO THE LONDON MATHEMATICAL SOCIETY
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The Maximum Degree&DiameterBounded Subgraph in the Mesh
, 2012
"... The problem of finding the largest connected subgraph of a given undirected host graph, subject to constraints on the maximum degree ∆ and the diameter D, was introduced in [1], as a generalization of the DegreeDiameter Problem. A case of special interest is when the host graph is a common parallel ..."
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The problem of finding the largest connected subgraph of a given undirected host graph, subject to constraints on the maximum degree ∆ and the diameter D, was introduced in [1], as a generalization of the DegreeDiameter Problem. A case of special interest is when the host graph is a common parallel architecture. Here we discuss the case when the host graph is a kdimensional mesh. We provide some general bounds for the order of the largest subgraph in arbitrary dimension k, and for the particular cases of k = 3, ∆ = 4 and k = 2, ∆ = 3, we give constructions that result in sharper lower bounds.
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:
doi:10.1093/imrn/rnt102 Counting Lattice Points and OMinimal Structures
"... Let Λ be a lattice in Rn, and let Z ⊆Rm+n be a definable family in an Ominimal structure over R. We give sharp estimates for the number of lattice points in the fibers ZT = {x∈ Rn: (T, x) ∈ Z}. Along the way, we show that for any subspace Σ ⊆Rn of dimension j> 0 the jvolume of the orthogonal p ..."
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Let Λ be a lattice in Rn, and let Z ⊆Rm+n be a definable family in an Ominimal structure over R. We give sharp estimates for the number of lattice points in the fibers ZT = {x∈ Rn: (T, x) ∈ Z}. Along the way, we show that for any subspace Σ ⊆Rn of dimension j> 0 the jvolume of the orthogonal projection of ZT to Σ is, up to a constant depending only on the family Z, bounded by the maximal jdimensional volume of the orthogonal projections to the jdimensional coordinate subspaces. 1