Linear relations with conjugates of a Salem number

In this paper we consider linear relations with conjugates of a Salem number $\alpha$. We show that every such a relation arises from a linear relation between conjugates of the corresponding totally real algebraic integer $\alpha+1/\alpha$. It is also shown that the smallest degree of a Salem number with a nontrivial relation between its conjugates is $8$, whereas the smallest length of a nontrivial linear relation between the conjugates of a Salem number is $6$.Comment: v1, 12 page

Salem numbers and Pisot numbers via interlacing

We present a general construction of Salem numbers via rational functions whose zeros and poles mostly lie on the unit circle and satisfy an interlacing condition. This extends and unifies earlier work. We then consider the "obvious" limit points of the set of Salem numbers produced by our theorems, and show that these are all Pisot numbers, in support of a conjecture of Boyd. We then show that all Pisot numbers arise in this way. Combining this with a theorem of Boyd, we show that all Salem numbers are produced via an interlacing construction.Comment: 21 pages, 5 figures, updated in response to reviewer comment

A new construction of Salem polynomials

An earlier result of the author on the zeros of reciprocal polynomials is applied to give a new construction of Salem number

Explicit Methods in Number Theory

[no abstract available

Cyclotomic points on curves

We show that a plane algebraic curve f = 0over the complex numbers has on it either at most 22V (f) points whose coordinates are both roots of unity, or innitely many such points. Here V (f) is the area of the Newton polytope of f: We present an algorithm for nding all these points

Dynamics on K3 Surfaces: Salem Numbers and Siegel Disks

This paper presents the first examples of K3 surface automorphisms $f : X \rightarrow X$ with Siegel disks (domains on which f acts by an irrational rotation). The set of such examples is countable, and the surface $X$ must be non-projective to carry a Siegel disk. These automorphisms are synthesized from Salem numbers of degree 22 and trace −1, which play the role of the leading eigenvalue for $f*|H^2(X)$. The construction uses the Torelli theorem, the Atiyah-Bott fixed-point theorem and results from transcendence theory.Mathematic