Approximating real-rooted and stable polynomials, with combinatorial applications

Let \( p(x) = a_0 + a_1x + \dots + a_nx^n \) be a polynomial with all roots real and satisfying \( x \leq -\delta \) for some \( 0 < \delta < 1 \). We show that for any \( 0 < \epsilon 0 \). As a corollary, we show that if \( m_k(G) \) is the number of matchings with \( k \) edges in a graph \( G \), then for any \( 0 < \epsilon 0 \) is an absolute constant. We prove a similar result for polynomials with complex roots satisfying \( \Re z \leq -\delta \) and apply it to estimate the number of unbranched subgraphs of \( G \).