366 research outputs found
The distribution of Mahler's measures of reciprocal polynomials
We study the distribution of Mahler's measures of reciprocal polynomials with
complex coefficients and bounded even degree. We discover that the distribution
function associated to Mahler's measure restricted to monic reciprocal
polynomials is a reciprocal (or anti-reciprocal) Laurent polynomial on
[1,\infty) and identically zero on [0,1). Moreover, the coefficients of this
Laurent polynomial are rational numbers times a power of \pi. We are led to
this discovery by the computation of the Mellin transform of the distribution
function. This Mellin transform is an even (or odd) rational function with
poles at small integers and residues that are rational numbers times a power of
\pi. We also use this Mellin transform to show that the volume of the set of
reciprocal polynomials with complex coefficients, bounded degree and Mahler's
measure less than or equal to one is a rational number times a power of \pi.Comment: 13 pages. To be published in Int. J. Math. Math. Sc
The Ginibre ensemble of real random matrices and its scaling limits
We give a closed form for the correlation functions of ensembles of a class
of asymmetric real matrices in terms of the Pfaffian of an antisymmetric matrix
formed from a matrix kernel associated to the ensemble. We apply
this result to the real Ginibre ensemble and compute the bulk and edge scaling
limits of its correlation functions as the size of the matrices becomes large.Comment: 47 pages, 8 figure
Equidistribution of Algebraic Numbers of Norm One in Quadratic Number Fields
Given a fixed quadratic extension K of Q, we consider the distribution of
elements in K of norm 1 (denoted N). When K is an imaginary quadratic
extension, N is naturally embedded in the unit circle in C and we show that it
is equidistributed with respect to inclusion as ordered by the absolute Weil
height. By Hilbert's Theorem 90, an element in N can be written as
\alpha/\bar{\alpha} for some \alpha \in O_K, which yields another ordering of
\mathcal N given by the minimal norm of the associated algebraic integers. When
K is imaginary we also show that N is equidistributed in the unit circle under
this norm ordering. When K is a real quadratic extension, we show that N is
equidistributed with respect to norm, under the map \beta \mapsto \log| \beta |
\bmod{\log | \epsilon^2 |} where \epsilon is a fundamental unit of O_K.Comment: 19 pages, 2 figures, comments welcome
The reciprocal Mahler ensembles of random polynomials
We consider the roots of uniformly chosen complex and real reciprocal polynomials of degree N whose Mahler measure is bounded by a constant. After a change of variables, this reduces to a generalization of Ginibre’s complex and real ensembles of random matrices where the weight function (on the eigenvalues of the matrices) is replaced by the exponentiated equilibrium potential of the interval [−2,2] on the real axis in the complex plane. In the complex (real) case, the random roots form a determinantal (Pfaffian) point process, and in both cases, the empirical measure on roots converges weakly to the arcsine distribution supported on [−2,2]. Outside this region, the kernels converge without scaling, implying among other things that there is a positive expected number of outliers away from [−2,2]. These kernels as well as the scaling limits for the kernels in the bulk (−2,2) and at the endpoints {−2,2} are presented. These kernels appear to be new, and we compare their behavior with related kernels which arise from the (non-reciprocal) Mahler measure ensemble of random polynomials as well as the classical Sine and Bessel kernels
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