Seventy years of Salem numbers

Smyth, Chris

doi:10.1112/blms/bdv027

Cited by 56 publications

(57 citation statements)

References 103 publications

(121 reference statements)

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“…For an example where (H4) fails, let G = G 4 m , Γ = {1} and V = G. We choose f = σ ∈ End(G) ∼ = M 4 (Z) to be the companion matrix of the minimal polynomial g of a Salem number α > 1 of degree 4 (e.g. g = x 4 − 3x 3 + 3x 2 − 3x + 1 [46]). Then deg(σ n − 1) = | det(σ n − 1)| and if β ∈ C is a zero of g with absolute value 1, then α and αβ are distinct dominant roots of the linear recurrent sequence (deg(σ n − 1)) n 1 .…”

Section: Hypothesis (H2)mentioning

confidence: 99%

Dynamically affine maps in positive characteristic

Byszewski¹,

Cornelissen²,

Houben³

2020

Dynamics: Topology and Numbers

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We study fixed points of iterates of dynamically affine maps (a generalisation of Lattès maps) over algebraically closed fields of positive characteristic p. We present and study certain hypotheses that imply a dichotomy for the Artin-Mazur zeta function of the dynamical system: it is either rational or non-holonomic, depending on specific characteristics of the map. We also study the algebraicity of the so-called tame zeta function, the generating function for periodic points of order coprime to p. We then verify these hypotheses for dynamically affine maps on the projective line, generalising previous work of Bridy, and, in arbitrary dimension, for maps on Kummer varieties arising from multiplication by integers on abelian varieties.

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Section: Hypothesis (H2)mentioning

confidence: 99%

Dynamically affine maps in positive characteristic

Byszewski¹,

Cornelissen²,

Houben³

2020

Dynamics: Topology and Numbers

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“…This set is related to the notion of Salem numbers. We want to quickly recall this notion here, for a general introduction we refer to [Sm15]. By definition, a Salem number is a real algebraic integer r > 0, all of whose Galois conjugates different from r lie in the closed unit disc |z| ≤ 1, with at least one on its boundary |z| = 1.…”

Section: Odd-dimensional Groups With Two-dimensional Centrementioning

confidence: 99%

Existence of Cocompact Lattices in Lie Groups With a Bi-invariant Metric of Index 2

Kath

2020

International Mathematics Research Notices

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We study the existence of cocompact lattices in Lie groups with bi-invariant metric of signature (2, n − 2). We assume in addition that the Lie groups under consideration are simply-connected, indecomposable and solvable. Then their centre is one-or two-dimensional. In both cases, a parametrisation of the set of such Lie groups is known. We give a necessary and sufficient condition for the existence of a lattice in terms of these parameters. For groups with one-dimensional centre this problem is related to Salem numbers. MSC2010: 53C50, 22E40, 57S30

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“…A breakthrough towards the solution of Mahler's problem was given by Smyth in [63]: The Mahler problem is, however, still open for SR polynomials. A monic integer SR polynomial with exactly two (real and positive) zeros (say, ζ and 1/ζ) not lying on S is called a Salem polynomial [62,64]. It can be shown that a Pisot polynomial with at least one zero on S is also a Salem polynomial.…”

Section: Polynomials With Small Mahler Measurementioning

confidence: 99%

Polynomials with Symmetric Zeros

Vieira

2019

Polynomials - Theory and Application

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Polynomials whose zeros are symmetric either to the real line or to the unit circle are very important in mathematics and physics. We can classify them into three main classes: the self-conjugate polynomials, whose zeros are symmetric to the real line; the self-inversive polynomials, whose zeros are symmetric to the unit circle; and the self-reciprocal polynomials, whose zeros are symmetric by an inversion with respect to the unit circle followed by a reflection in the real line. Real selfreciprocal polynomials are simultaneously self-conjugate and self-inversive so that their zeros are symmetric to both the real line and the unit circle. In this survey, we present a short review of these polynomials, focusing on the distribution of their zeros.

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Seventy years of Salem numbers

Abstract: I survey results about, and recent applications of, Salem numbers.

Cited by 56 publications

References 103 publications

Dynamically affine maps in positive characteristic

Dynamically affine maps in positive characteristic

Existence of Cocompact Lattices in Lie Groups With a Bi-invariant Metric of Index 2

Polynomials with Symmetric Zeros

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