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1. CJM Online first
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 2008 (vol 60 pp. 1149)
Conjugate Reciprocal Polynomials with All Roots on the Unit Circle We study the geometry, topology and Lebesgue measure of the set of
monic conjugate reciprocal polynomials of fixed degree with all
roots on the unit circle. The set of such polynomials of degree $N$
is naturally associated to a subset of $\R^{N-1}$. We calculate
the volume of this set, prove the set is homeomorphic to the $N-1$
ball and that its isometry group is isomorphic to the dihedral
group of order $2N$.
Categories:11C08, 28A75, 15A52, 54H10, 58D19 |
3. CJM 2005 (vol 57 pp. 1080)
The Gelfond--Schnirelman Method in Prime Number Theory The original Gelfond--Schnirelman method, proposed in 1936, uses
polynomials with integer coefficients and small norms on $[0,1]$
to give a Chebyshev-type lower bound in prime number theory. We
study a generalization of this method for polynomials in many
variables. Our main result is a lower bound for the integral of
Chebyshev's $\psi$-function, expressed in terms of the weighted
capacity. This extends previous work of Nair and Chudnovsky, and
connects the subject to the potential theory with external fields
generated by polynomial-type weights. We also solve the
corresponding potential theoretic problem, by finding the extremal
measure and its support.
Keywords:distribution of prime numbers, polynomials, integer, coefficients, weighted transfinite diameter, weighted capacity, potentials Categories:11N05, 31A15, 11C08 |