绝对根分离

随机多项式根间隔的极限律

Limit law for root separation in random polynomials

本文研究系数独立同分布的随机多项式根的间隔，证明其经 $n^{-5/4}$ 归一化后收敛到非齐次泊松点过程，并由此得到最小根间隔的非平凡极限律。

Extremal problems for polynomials with real roots

We consider polynomials of degree $d$ with only real roots and a fixed value of discriminant, and study the problem of minimizing the absolute value of polynomials at a fixed point off the real line. There are two explicit families of polynomials that turn out to be extremal in terms of this problem. The first family has a particularly simple expression as a linear combination of $d$-th powers of two linear functions. Moreover, if the value of the discriminant is not too small, then the roots of the extremal polynomial and the smallest absolute value in question can be found explicitly. The se…

Perturbation of complex polynomials and normal operators

We study the regularity of the roots of complex monic polynomials $P(t)$ of fixed degree depending smoothly on a real parameter $t$. We prove that each continuous parameterization of the roots of a generic $C^\infty$ curve $P(t)$ (which always exists) is locally absolutely continuous. Generic means that no two of the continuously chosen roots meet of infinite order of flatness. Simple examples show that one cannot expect a better regularity than absolute continuity. This result will follow from the proposition that for any $t_0$ there exists a positive integer $N$ such that $t \mapsto P(t_0\pm…

Stability of roots of polynomials under linear combinations of derivatives

Let $T=α_0 I + α_1 D + ...+α_n D^n$, where $D$ is the differentiation operator and $α_0\not= 0$, and let $f$ be a square-free polynomial with large minimum root separation. We prove that the roots of $Tf$ are close to the roots of $f$ translated by $-α_1/α_0$.

Regularity of roots of polynomials

We show that smooth curves of monic complex polynomials $P_a (Z)=Z^n+\sum_{j=1}^n a_j Z^{n-j}$, $a_j : I \to \mathbb C$ with $I \subset \mathbb R$ a compact interval, have absolutely continuous roots in a uniform way. More precisely, there exists a positive integer $k$ and a rational number $p >1$, both depending only on the degree $n$, such that if $a_j \in C^{k}$ then any continuous choice of roots of $P_a$ is absolutely continuous with derivatives in $L^q$ for all $1 \le q < p$, in a uniform way with respect to $\max_j\|a_j\|_{C^k}$. The uniformity allows us to deduce also a multiparameter …

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 $x \leq -δ$ 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 $k \leq c \sqrtΔ \ln (v /ε)$…