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Search: MSC category 30C15 ( Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) {For algebraic theory, see 12D10; for real methods, see 26C10} )

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1. CJM 2008 (vol 60 pp. 958)

Chen, Yichao
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)

Cartwright, Donald I.; Steger, Tim
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)

Lubinsky, D. S.
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)

Stahl, Saul
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

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