CMC Faculty Publications and ResearchCopyright (c) 2016 Claremont Colleges All rights reserved.
http://scholarship.claremont.edu/cmc_fac_pub
Recent documents in CMC Faculty Publications and Researchen-usSat, 10 Dec 2016 01:48:02 PST3600On Arithmetic Lattices in the Plane
http://scholarship.claremont.edu/cmc_fac_pub/450
http://scholarship.claremont.edu/cmc_fac_pub/450Fri, 09 Dec 2016 14:07:46 PST
We investigate similarity classes of arithmetic lattices in the plane. We introduce a natural height function on the set of such similarity classes, and give asymptotic estimates on the number of all arithmetic similarity classes, semi-stable arithmetic similarity classes, and well-rounded arithmetic similarity classes of bounded height as the bound tends to infinity. We also briefly discuss some properties of the j-invariant corresponding to similarity classes of planar lattices.
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Lenny Fukshansky et al.Lattice Theory and Toeplitz Determinants
http://scholarship.claremont.edu/cmc_fac_pub/449
http://scholarship.claremont.edu/cmc_fac_pub/449Fri, 09 Dec 2016 14:07:37 PST
This is a survey of our recent joint investigations of lattices that are generated by finite Abelian groups. In the case of cyclic groups, the volume of a fundamental domain of such a lattice is a perturbed Toeplitz determinant with a simple Fisher-Hartwig symbol. For general groups, the situation is more complicated, but it can still be tackled by pure matrix theory. Our main result on the lattices under consideration states that they always have a basis of minimal vectors, while our results in the other direction concern exact and asymptotic formulas for perturbed Toeplitz determinants. The survey is a slightly modified version of the talk given by the first author at the Humboldt Kolleg and the IWOTA in Tbilisi in 2015. It is mainly for operator theorists and therefore also contains an introduction to the basics of lattice theory.
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Albrecht Böttcher et al.Spherical 2-Designs and Lattices from Abelian Groups
http://scholarship.claremont.edu/cmc_fac_pub/448
http://scholarship.claremont.edu/cmc_fac_pub/448Fri, 09 Dec 2016 14:07:30 PST
We consider lattices generated by finite Abelian groups. The main result says that such a lattice is strongly eutactic, which means the normalized minimal vectors of the lattice form a spherical 2-design, if and only if the group is of odd order or if it is a power of the group of order 2. This result also yields a criterion for the appropriately normalized minimal vectors to constitute a uniform normalized tight frame.
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Albrecht Böttcher et al.On an Effective Variation of Kronecker's Approximation Theorem
http://scholarship.claremont.edu/cmc_fac_pub/447
http://scholarship.claremont.edu/cmc_fac_pub/447Fri, 09 Dec 2016 14:07:22 PST
Let Λ ⊂ Rn be an algebraic lattice, coming from a projective module over the ring of integers of a number field K. Let Z ⊂ Rn be the zero locus of a finite collection of polynomials such that Λ |⊂ Z or a finite union of proper full-rank sublattices of Λ. Let K1 be the number field generated over K by coordinates of vectors in Λ, and let L1, . . . , Lt be linear forms in n variables with algebraic coefficients satisfying an appropriate linear independence condition over K1. For each ε > 0 and a ∈ Rn, we prove the existence of a vector x ∈ Λ \ Z of explicitly bounded sup-norm such that ||Li(x) − ai|| < ε for each 1 ≤ i ≤ t, where || || stands for the distance to the nearest integer. The bound on sup-norm of x depends on Λ, K, Z, heights of linear forms, and ε. This presents a generalization of Kronecker’s approximation theorem, establishing an effective result on density of the image of Λ\Z under the linear forms L1, . . . , Lt in the t-torus Rt/Zt .
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Lenny FukshanskyStability of Ideal Lattices from Quadratic Number Fields
http://scholarship.claremont.edu/cmc_fac_pub/446
http://scholarship.claremont.edu/cmc_fac_pub/446Thu, 08 Dec 2016 12:55:15 PST
We study semi-stable ideal lattices coming from real quadratic number fields. Specifically, we demonstrate infinite families of semi-stable and unstable ideal lattices of trace type, establishing explicit conditions on the canonical basis of an ideal that ensure stability; in particular, our result implies that an ideal lattice of trace type coming from a real quadratic field is semi-stable with positive probability. We also briefly discuss the connection between stability and well-roundedness of Euclidean lattices.
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Lenny FukshanskyToeplitz Determinants with Perturbations in the Corners
http://scholarship.claremont.edu/cmc_fac_pub/445
http://scholarship.claremont.edu/cmc_fac_pub/445Thu, 08 Dec 2016 12:55:07 PST
The paper is devoted to exact and asymptotic formulas for the determinants of Toeplitz matrices with perturbations by blocks of fixed size in the four corners. If the norms of the inverses of the unperturbed matrices remain bounded as the matrix dimension goes to infinity, then standard perturbation theory yields asymptotic expressions for the perturbed determinants. This premise is not satisfied for matrices generated by so-called Fisher–Hartwig symbols. In that case we establish formulas for pure single Fisher–Hartwig singularities and for Hermitian matrices induced by general Fisher–Hartwig symbols.
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Albrecht Böttcher et al.On Lattices Generated by Finite Abelian Groups
http://scholarship.claremont.edu/cmc_fac_pub/444
http://scholarship.claremont.edu/cmc_fac_pub/444Thu, 08 Dec 2016 12:54:58 PST
This paper is devoted to the study of lattices generated by finite Abelian groups. Special species of such lattices arise in the exploration of elliptic curves over finite fields. In the case where the generating group is cyclic, they are also known as the Barnes lattices. It is shown that for every finite Abelian group with the exception of the cyclic group of order four these lattices have a basis of minimal vectors. Another result provides an improvement of a recent upper bound by M. Sha for the covering radius in the case of the Barnes lattices. Also discussed are properties of the automorphism groups of these lattices.
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Albrecht Böttcher et al.Permutation Invariant Lattices
http://scholarship.claremont.edu/cmc_fac_pub/443
http://scholarship.claremont.edu/cmc_fac_pub/443Thu, 08 Dec 2016 12:54:50 PST
We say that a Euclidean lattice in Rn is permutation invariant if its automorphism group has non-trivial intersection with the symmetric group Sn, i.e., if the lattice is closed under the action of some non-identity elements of Sn. Given a fixed element τ ∈ Sn, we study properties of the set of all lattices closed under the action of τ: we call such lattices τ-invariant. These lattices naturally generalize cyclic lattices introduced by Micciancio in [8, 9], which we previously studied in [1]. Continuing our investigation, we discuss some basic properties of permutation invariant lattices, in particular proving that the subset of well-rounded lattices in the set of all τ-invariant lattices in Rn has positive co-dimension (and hence comprises zero proportion) for all τ different from an n-cycle.
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Lenny Fukshansky et al.Height Bounds on Zeros of Quadratic Forms Over Q-bar
http://scholarship.claremont.edu/cmc_fac_pub/442
http://scholarship.claremont.edu/cmc_fac_pub/442Thu, 08 Dec 2016 12:54:42 PST
In this paper we establish three results on small-height zeros of quadratic polynomials over Q. For a single quadratic form in N ≥ 2 variables on a subspace of Q N , we prove an upper bound on the height of a smallest nontrivial zero outside of an algebraic set under the assumption that such a zero exists. For a system of k quadratic forms on an L-dimensional subspace of Q N , N ≥ L ≥ k(k+1) 2 + 1, we prove existence of a nontrivial simultaneous small-height zero. For a system of one or two inhomogeneous quadratic and m linear polynomials in N ≥ m + 4 variables, we obtain upper bounds on the height of a smallest simultaneous zero, if such a zero exists. Our investigation extends previous results on small zeros of quadratic forms, including Cassels’ theorem and its various generalizations and contributes to the literature of so-called “absolute” Diophantine results with respect to height. All bounds on height are explicit.
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Lenny FukshanskyTotally Isotropic Subspaces of Small Height in Quadratic Spaces
http://scholarship.claremont.edu/cmc_fac_pub/441
http://scholarship.claremont.edu/cmc_fac_pub/441Thu, 08 Dec 2016 12:54:34 PST
Let K be a global field or Q, F a nonzero quadratic form on KN , N ≥ 2, and V a subspace of KN . We prove the existence of an infinite collection of finite families of small-height maximal totally isotropic subspaces of (V, F) such that each such family spans V as a K-vector space. This result generalizes and extends a well known theorem of J. Vaaler [16] and further contributes to the effective study of quadratic forms via height in the general spirit of Cassels’ theorem on small zeros of quadratic forms. All bounds on height are explicit.
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Wai Kiu Chan et al.Lattices from Hermitian Function Fields
http://scholarship.claremont.edu/cmc_fac_pub/440
http://scholarship.claremont.edu/cmc_fac_pub/440Thu, 08 Dec 2016 12:54:27 PST
We consider the well-known Rosenbloom–Tsfasman function field lattices in the special case of Hermitian function fields. We show that in this case the resulting lattices are generated by their minimal vectors, provide an estimate on the total number of minimal vectors, and derive properties of the automorphism groups of these lattices. Our study continues previous investigations of lattices coming from elliptic curves and finite Abelian groups. The lattices we are faced with here are more subtle than those considered previously, and the proofs of the main results require the replacement of the existing linear algebra approaches by deep results of Gerhard Hiss on the factorization of functions with particular divisor support into lines and their inverses.
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Albrecht Böttcher et al.