Expand all Collapse all | Results 1 - 5 of 5 |
1. CJM 2008 (vol 60 pp. 958)
A Note on a Conjecture of S. Stahl S. Stahl (Canad. J. Math. \textbf{49}(1997), no. 3, 617--640)
conjectured that the zeros of genus polynomial are real.
L. Liu and Y. Wang disproved this conjecture on the basis
of Example 6.7. In this note, it is pointed out
that there is an error in this example and a new generating matrix
and initial vector are provided.
Keywords:genus polynomial, zeros, real Categories:05C10, 05A15, 30C15, 26C10 |
2. CJM 2008 (vol 60 pp. 960)
3. CJM 2002 (vol 54 pp. 239)
Elementary Symmetric Polynomials in Numbers of Modulus $1$ We describe the set of numbers $\sigma_k(z_1,\ldots,z_{n+1})$, where
$z_1,\ldots,z_{n+1}$ are complex numbers of modulus $1$ for which
$z_1z_2\cdots z_{n+1}=1$, and $\sigma_k$ denotes the $k$-th
elementary symmetric polynomial. Consequently, we give sharp
constraints on the coefficients of a complex polynomial all of whose
roots are of the same modulus. Another application is the calculation
of the spectrum of certain adjacency operators arising naturally
on a building of type ${\tilde A}_n$.
Categories:05E05, 33C45, 30C15, 51E24 |
4. CJM 2000 (vol 52 pp. 815)
On the Maximum and Minimum Modulus of Rational Functions We show that if $m$, $n\geq 0$, $\lambda >1$, and $R$ is a rational function
with numerator, denominator of degree $\leq m$, $n$, respectively, then there
exists a set $\mathcal{S}\subset [0,1] $ of linear measure $\geq
\frac{1}{4}\exp (-\frac{13}{\log \lambda })$ such that for $r\in
\mathcal{S}$,
\[
\max_{|z| =r}| R(z)| / \min_{|z| =r} | R(z) |\leq \lambda ^{m+n}.
\]
Here, one may not replace $\frac{1}{4}\exp ( -\frac{13}{\log \lambda })$
by $\exp (-\frac{2-\varepsilon }{\log \lambda })$, for any $\varepsilon >0$.
As our motivating application, we prove a convergence result for diagonal
Pad\'{e} approximants for functions meromorphic in the unit ball.
Categories:30E10, 30C15, 31A15, 41A21 |
5. CJM 1997 (vol 49 pp. 617)
On the zeros of some genus polynomials In the genus polynomial of the graph $G$, the coefficient of $x^k$
is the number of distinct embeddings of the graph $G$ on the
oriented surface of genus $k$. It is shown that for several
infinite families of graphs all the zeros of the genus polynomial
are real and negative. This implies that their coefficients, which
constitute the genus distribution of the graph, are log concave and
therefore also unimodal. The geometric distribution of the zeros
of some of these polynomials is also investigated and some new
genus polynomials are presented.
Categories:05C10, 05A15, 30C15, 26C10 |