The coordinator polynomial of some cyclotomic lattices
Frédéric Patras,Patrick Solé +1 more
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This work investigates the special case when L is the ring of integers of the cyclotomic field of order m and S is the corresponding set of unit roots, and compute the coordinator polynomial explicitly when m = p and m = 2p, with p an odd prime.Abstract:
The coordinator polynomial of a lattice L is the numerator of its growth series as an abelian group, w.r.t, to a given set of generators S. We investigate the special case when L is the ring of integers of the cyclotomic field of order m and S is the corresponding set of unit roots. We compute it explicitly when m = p and m = 2p, with p an odd prime. This confirms, for small p, a conjecture of Parker. Our approach is geometric and is grounded in the theory of Ehrhart polynomials.read more
Citations
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Cyclotomic Polytopes and Growth Series of Cyclotomic Lattices
Matthias Beck,Serkan Hosten +1 more
TL;DR: In this paper, the authors investigated the coordination sequence of the cyclotomic lattice and proved several conjectures by Parker regarding the structure of the rational generating function of coordination sequence; this structure depends on the prime factorization of the root of unity.
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Cyclotomic Polytopes and Growth Series of Cyclotomic Lattices
Matthias Beck,Serkan Hosten +1 more
TL;DR: In this paper, the authors investigated the coordination sequence of the cyclotomic lattice and proved several conjectures by Parker regarding the structure of the rational generating function of coordination sequence; this structure depends on the prime factorization of the root of unity.
Journal ArticleDOI
Analytic combinatorics of coordination numbers of cubic lattices
Huyile Liang,Yanni Pei,Yi Wang +2 more
TL;DR: In this paper , the authors investigate coordination numbers of the cubic lattices with emphases on their analytic behaviors, including the total positivity of the coordination matrices, the distribution of zero coefficients, the asymptotic normality of the coefficients of the coordinates, the log-concavity and the logconvexity of coordination numbers.
References
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Book
Groupes et algèbres de Lie
TL;DR: Les Elements de mathematique de Nicolas Bourbaki ont pour objet une presentation rigoureuse, systematique et sans prerequis des mathematiques depuis leurs fondements as mentioned in this paper.
Book
Introduction to Cyclotomic Fields
TL;DR: In this paper, Dirichlet characters were used to construct p-adic L-functions and Bernoulli numbers, which are then used to define the class number formula.
Book ChapterDOI
Cohomologie des groupes discrets
TL;DR: In this article, Bourbaki et al. implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.php).
Journal ArticleDOI
Low-dimensional lattices. vii. coordination sequences
John H. Conway,Neil J. A. Sloane +1 more
TL;DR: The coordination sequence S(n) of a lattice or net gives the number of nodes that are n bonds away from a given node as discussed by the authors, where S(1) is the familiar coordination number.