Quadratic cyclic sequences
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Quadratic cyclic sequences Paul Baird1
· Ali Fardoun1 · Zeina Ghazo Hanna1
Received: 4 September 2019 / Accepted: 25 July 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract We explore relations between cyclic sequences determined by a quadratic difference relation, cyclotomic polynomials, Eulerian digraphs and walks in the plane. These walks correspond to closed paths for which at each step one must turn either left or right through a fixed angle. In the case when this angle is 2π/n, then non-symmetric phenomena occurs for n ≥ 12. Examples arise from algebraic integers of modulus one which are not n’th roots of unity. Keywords Cyclic sequence · Quadratic difference relation · Cyclotomic polynomial · Digraph · Eulerian digraph · Planar lattice · Planar walk Mathematics Subject Classification 11B83 · 52C05 · 82B41
Communicated by Ilse Fischer. The authors thank Olivier Rahavandrainy and Benoit Saussol for helpful comments in respect of this work. They also thank the two referees whose careful reading of the paper have helped to improve its presentation.
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Paul Baird [email protected] Ali Fardoun [email protected] Zeina Ghazo Hanna [email protected]
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Laboratoire de Mathématiques de Bretagne Atlantique UMR 6205, Université de Brest, 6 Av. Victor Le Gorgeu – CS 93837, 29238 Brest Cedex, France
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1 Introduction For a given integer N ≥ 2, define a quadratic cyclic sequence (QCS) of order N to be a function ϕ : Z/N Z → C satisfying the quadratic difference relation 2 2 γ 2ϕ( j) − ϕ( j − 1) − ϕ( j + 1) = ϕ( j) − ϕ( j − 1) 2 2 + ϕ( j) − ϕ( j + 1) ∀ j ∈ Z/N Z
(1)
for some real number γ where j ± 1 are calculated modulo N . If we define the increment u j = ϕ( j + 1) − ϕ( j), then the above equation reads γ (u j − u j−1 )2 − u j 2 − u j−1 2 = 0, 2
(2)
which affirms the vanishing of a linear combination of the elementary symmetric quadratic polynomials u j 2 + u j−1 2 and u j u j−1 in the two variables u j and u j−1 . The more general equation, 2 2 γ (x) ϕ(y) − ϕ(x) ϕ(x) − ϕ(y) , = d(x) y∼x y∼x defined on an arbitrary simple graph, where d(x) is the degree at vertex x and y ∼ x means x and y are connected by an edge, arises in studies of projections to the plane of regular polytopes [4] and invariant spacial frameworks [1]. Equation (1) corresponds to the special case of a cyclic graph. The main focus of this article is to explore the relation between solutions of (1) and closed polygonal walks in the plane with the property that at each step, one must turn either left or right by a fixed given angle, called the turning angle. Such a walk determines a star polygon with sides of equal length and with exterior angle ±θ for some fixed θ (the turning angle). In the case when the turning angle is a rational multiple of 2π : θ = 2mπ/n (m, n coprime integers with m < n), we address two fundamental questions: (i) When n is even, must each step have a parallel counterpart oriented in the opposite direction somewhere along the path?
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