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Search: MSC category 11R06 ( PV-numbers and generalizations; other special algebraic numbers; Mahler measure )

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1. CJM Online first

Borwein, Peter; Choi, Stephen; Ferguson, Ron; Jankauskas, Jonas
On Littlewood Polynomials with Prescribed Number of Zeros Inside the Unit Disk
We investigate the numbers of complex zeros of Littlewood polynomials $p(z)$ (polynomials with coefficients $\{-1, 1\}$) inside or on the unit circle $|z|=1$, denoted by $N(p)$ and $U(p)$, respectively. Two types of Littlewood polynomials are considered: Littlewood polynomials with one sign change in the sequence of coefficients and Littlewood polynomials with one negative coefficient. We obtain explicit formulas for $N(p)$, $U(p)$ for polynomials $p(z)$ of these types. We show that, if $n+1$ is a prime number, then for each integer $k$, $0 \leq k \leq n-1$, there exists a Littlewood polynomial $p(z)$ of degree $n$ with $N(p)=k$ and $U(p)=0$. Furthermore, we describe some cases when the ratios $N(p)/n$ and $U(p)/n$ have limits as $n \to \infty$ and find the corresponding limit values.

Keywords:Littlewood polynomials, zeros, complex roots
Categories:11R06, 11R09, 11C08

2. CJM 2012 (vol 64 pp. 254)

Bell, Jason P.; Hare, Kevin G.
Corrigendum to ``On $\mathbb{Z}$-modules of Algebraic Integers''
We fix a mistake in the proof of Theorem 1.6 in the paper in the title.

Keywords:Pisot numbers, algebraic integers, number rings, Schmidt subspace theorem
Categories:11R04, 11R06

3. CJM 2011 (vol 64 pp. 345)

McKee, James; Smyth, Chris
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 produce all Salem numbers via an interlacing construction.

Keywords:Salem numbers, Pisot numbers
Category:11R06

4. CJM 2010 (vol 62 pp. 543)

Hare, Kevin G.
More Variations on the Sierpiński Sieve
This paper answers a question of Broomhead, Montaldi and Sidorov about the existence of gaskets of a particular type related to the Sierpiński sieve. These gaskets are given by iterated function systems that do not satisfy the open set condition. We use the methods of Ngai and Wang to compute the dimension of these gaskets.

Categories:28A80, 28A78, 11R06

5. CJM 2009 (vol 61 pp. 264)

Bell, J. P.; Hare, K. G.
On $\BbZ$-Modules of Algebraic Integers
Let $q$ be an algebraic integer of degree $d \geq 2$. Consider the rank of the multiplicative subgroup of $\BbC^*$ generated by the conjugates of $q$. We say $q$ is of {\em full rank} if either the rank is $d-1$ and $q$ has norm $\pm 1$, or the rank is $d$. In this paper we study some properties of $\BbZ[q]$ where $q$ is an algebraic integer of full rank. The special cases of when $q$ is a Pisot number and when $q$ is a Pisot-cyclotomic number are also studied. There are four main results. \begin{compactenum}[\rm(1)] \item If $q$ is an algebraic integer of full rank and $n$ is a fixed positive integer, then there are only finitely many $m$ such that $\disc\left(\BbZ[q^m]\right)=\disc\left(\BbZ[q^n]\right)$. \item If $q$ and $r$ are algebraic integers of degree $d$ of full rank and $\BbZ[q^n] = \BbZ[r^n]$ for infinitely many $n$, then either $q = \omega r'$ or $q={\rm Norm}(r)^{2/d}\omega/r'$, where $r'$ is some conjugate of $r$ and $\omega$ is some root of unity. \item Let $r$ be an algebraic integer of degree at most $3$. Then there are at most $40$ Pisot numbers $q$ such that $\BbZ[q] = \BbZ[r]$. \item There are only finitely many Pisot-cyclotomic numbers of any fixed order. \end{compactenum}

Keywords:algebraic integers, Pisot numbers, full rank, discriminant
Categories:11R04, 11R06

6. CJM 2002 (vol 54 pp. 468)

Boyd, David W.; Rodriguez-Villegas, Fernando
Mahler's Measure and the Dilogarithm (I)
An explicit formula is derived for the logarithmic Mahler measure $m(P)$ of $P(x,y) = p(x)y - q(x)$, where $p(x)$ and $q(x)$ are cyclotomic. This is used to find many examples of such polynomials for which $m(P)$ is rationally related to the Dedekind zeta value $\zeta_F (2)$ for certain quadratic and quartic fields.

Categories:11G40, 11R06, 11Y35

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