Approximating real-rooted and stable polynomials, with combinatorial applications
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Let p(x)=a_0 + a_1 x + ... + a_n x^n be a polynomial with all roots real and satisfying x ≤ -δ for some 0<δ <1. We show that for any 0 < ϵ <1, the value of p(1) is determined within relative error ϵ by the coefficients a_k with k ≤c √(δ)n ϵ√(δ) 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 < ϵ < 1, the total number M(G)=m_0(G)+m_1(G) + ... of matchings is determined within relative error ϵ by the numbers m_k(G) with k ≤ c √(Δ) (v /ϵ), where Δ 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 z ≤ -δ and apply it to estimate the number of unbranched subgraphs of G.
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