Approximating real-rooted and stable polynomials, with combinatorial applications

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 $G$.