Approximating realrooted and stable polynomials, with combinatorial applications
Abstract
Let $p(x)=a_0 + a_1 x + \ldots + a_n x^n$ be a polynomial with all roots real and satisfying $x \leq \delta$ for some $0<\delta <1$. We show that for any $0 < \epsilon <1$, the value of $p(1)$ is determined within relative error $\epsilon$ by the coefficients $a_k$ with $k \leq {c \over \sqrt{\delta}} \ln {n \over \epsilon \sqrt{ \delta}}$ for some absolute constant $c > 0$. Consequently, if $m_k(G)$ is the number of matchings with $k$ edges in a graph $G$, then for any $0 < \epsilon < 1$, the total number $M(G)=m_0(G)+m_1(G) + \ldots $ of matchings is determined within relative error $\epsilon$ by the numbers $m_k(G)$ with $k \leq c \sqrt{\Delta} \ln (v /\epsilon)$, where $\Delta$ is the largest degree of a vertex, $v$ is the number of vertices of $G$ and $c >0$ is an absolute constant. We prove a similar result for polynomials with complex roots satisfying $\Re\thinspace z \leq \delta$ and apply it to estimate the number of unbranched subgraphs of $G$.
 Publication:

arXiv eprints
 Pub Date:
 June 2018
 arXiv:
 arXiv:1806.07404
 Bibcode:
 2018arXiv180607404B
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Data Structures and Algorithms;
 Mathematics  Classical Analysis and ODEs;
 26C10;
 30C15;
 05A16;
 05C30;
 05C31
 EPrint:
 12 pages