arXiv 1806.074042018-06-19math.COcs.DSmath.CA

近似实根和稳定多项式，以及组合应用

Approximating real-rooted and stable polynomials, with combinatorial applications

Alexander Barvinok

备注：: 12 pages

摘要：

设$p(x)=a_0 + a_1 x + \ldots + a_n x^n$是一个多项式，其所有根都是实的，且对某些$0<δ<1$满足$x \leq -δ$.我们表明，对于任何$0 < ε<1$，$p(1)$的值是确定在相对误差$ε$的系数$a_k$与$k \leq {c \over \sqrtδ} \ln {n \over ε\sqrt{ δ}}$的一些绝对常数$c > 0$。因此，如果$m_k(G)$是图$G$中具有$k$边的匹配数，则对于任何$0 < ε< 1$，匹配的总数$M(G)=m_0(G)+m_1(G) + \ldots $在相对误差$ε$内由具有$k \leq c \sqrtΔ \ln (v /ε)$的$m_k(G)$的数目确定，其中$Δ$是顶点的最大度，$v$是$G$的顶点数，$c >0$是绝对常数。证明了满足$\Re\thinspace z \leq -δ$的复根多项式的一个类似结果，并将其应用于估计$G$的无分支子图的个数。

英文摘要：

Let $p(x)=a_0 + a_1 x + \ldots + a_n x^n$ be a polynomial with all roots real and satisfying $x \leq -δ$ 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 $k \leq {c \over \sqrtδ} \ln {n \over ε\sqrt{ δ}}$ 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) + \ldots $ of matchings is determined within relative error $ε$ by the numbers $m_k(G)$ with $k \leq c \sqrtΔ \ln (v /ε)$, 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 $\Re\thinspace z \leq -δ$ and apply it to estimate the number of unbranched subgraphs of $G$.

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