Analytic theory of polynomials

TL;DR: In this paper, the authors give two possible generalizations to rational maps, both of which are Mobius invariant, and prove a version with a weaker constant, in parallel to the situation for Smale's mean value conjecture.

Abstract: For a graph $G=(V,E)$, $k\in \mathbb{N}$, and a complex number $w$ the partition function of the univariate Potts model is defined as \[ {\bf Z}(G;k,w):=\sum_{\phi:V\to [k]}\prod_{\substack{uv\in E \\ \phi(u)=\phi(v)}}w, \] where $[k]:=\{1,\ldots,k\}$. In this paper we give zero-free regions for the partition function of the anti-ferromagnetic Potts model on bounded degree graphs. In particular we show that for any $\Delta\in \mathbb{N}$ and any $k\geq e\Delta+1$, there exists an open set $U$ in the complex plane that contains the interval $[0,1)$ such that ${\bf Z}(G;k,w) eq 0$ for any $w\in U$ and any graph $G$ of maximum degree at most $\Delta$. (Here $e$ denotes the base of the natural logarithm.) For small values of $\Delta$ we are able to give better results. As an application of our results we obtain improved bounds on $k$ for the existence of deterministic approximation algorithms for counting the number of proper $k$-colourings of graphs of small maximum degree.

TL;DR: In this paper, the authors investigate the distribution of zeros of entire functions which can be represented as the Fourier transforms of certain admissible kernels and bring to light the intimate connection between the Bochner-Khinchin-Mathias theory of positive definite kernels and the generalized real Laguerre inequalities.

TL;DR: In this paper, it was shown that the moduli of the zeros of a polynomial f(z) weakly majorize those of zf′(z), which refines the Gauss-Lucas Theorem.