On the Roots of a Hyperbolic Polynomial Pencil
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- 作者：Victor Katsnelson
- 关键词：Hyperbolic polynomial pencil ; Determinant representation ; Exponentially convex functions
- 刊名：Arnold Mathematical Journal
- 出版年：2016
- 出版时间：December 2016
- 年：2016
- 卷：2
- 期：4
- 页码：439-448
- 全文大小：462 KB
- 刊物类别：Mathematics, general; Algebraic Geometry; Mathematical Physics; Analysis; Dynamical Systems and Ergo
- 刊物主题：Mathematics, general; Algebraic Geometry; Mathematical Physics; Analysis; Dynamical Systems and Ergodic Theory; Combinatorics;
- 出版者：Springer International Publishing
- ISSN：2199-6806
- 卷排序：2
文摘
Let \(\nu _0(t),\nu _1(t),\ldots ,\nu _n(t)\) be the roots of the equation \(R(z)=t\), where R(z) is a rational function of the form $$\begin{aligned} R(z)=z-\sum \limits _{k=1}^n\frac{\alpha _k}{z-\mu _k}, \end{aligned}$$\(\mu _k\) are pairwise distinct real numbers, \(\alpha _k>0,\,1\le {}k\le {}n\). Then for each real \(\xi \), the function \(e^{\xi \nu _0(t)}+e^{\xi \nu _1(t)}+\,\cdots \,+e^{\xi \nu _n(t)}\) is exponentially convex on the interval \(-\infty<t<\infty \).KeywordsHyperbolic polynomial pencilDeterminant representationExponentially convex functionsMathematics Subject Classification11C9926C1026C1515A2242A821 Roots of the Equation \({R(z)=t}\) as Functions of tIn the present paper we discuss questions related to properties of roots of the equation$$\begin{aligned} R(z)=t \end{aligned}$$ (1.1)as functions of the parameter \(t\in \mathbb {C}\), where R is a rational function of the form$$\begin{aligned} R(z)=z-\sum \limits _{1\le k \le n}\frac{\alpha _{k}}{z-\mu _{k}}, \end{aligned}$$ (1.2)\(\mu _k\) are pairwise distinct real numbers, \(\alpha _k>0\), \(1\le k\le n\). We adhere to the enumeration agreement1$$\begin{aligned} \mu _1>\mu _2>\cdots >\mu _n. \end{aligned}$$ (1.3)The function R is representable in the form$$\begin{aligned} R(z)=\frac{P(z)}{Q(z)}, \end{aligned}$$ (1.4)where$$\begin{aligned}&Q(z)=(z-\mu _1)\cdot (z-\mu _2)\cdot \,\cdots \,\cdot (z-\mu _n),\end{aligned}$$ (1.5)$$\begin{aligned}&P(z)\mathop {=}\limits ^{\text {def}}R(z)\cdot {}Q(z) \end{aligned}$$ (1.6)are monic polynomials of degrees$$\begin{aligned} \text {deg}\,P=n+1,\quad \text {deg}\,Q=n. \end{aligned}$$ (1.7)Since \(P(\mu _k)=-\alpha _k Q^{\prime }(\mu _k)\not =0\), the polynomials P and Q have no common roots. Thus the ratio in the right hand side of (1.4) is irreducible. The Eq. (1.1) is equivalent to the equation$$\begin{aligned} P(z)-tQ(z)=0. \end{aligned}$$ (1.8)Since the polynomial \(P(z)-tQ(z)\) is of degree \(n+1\), the latter equation has \(n+1\) roots for each \(t\in \mathbb {C}\).The function R possess the property$$\begin{aligned} \text {Im}\,R(z)\big /\text {Im}\,z>0 \quad \ \text {if}\quad \text {Im}\,z\not = 0. \end{aligned}$$ (1.9)Therefore if \(\text {Im}\,t>0\), all roots of the equation (1.1), which is equivalent to the Eq. (1.8), are located in the half-plane \(\text {Im}\,z>0\). Some of these roots may be multiple.However if t is real, all roots of the Eq. (1.1) are real and simple, i.e. of multiplicity one. Thus for real t, the Eq. (1.1) has \(n+1\) pairwise distinct real roots \(\nu _k(t)\): \(\nu _0(t)>\nu _1(t)>\cdots>\nu _{n-1}(t)>\nu _n(t)\). Moreover for each real t, the poles \(\mu _k\) of the function R and the roots \(\nu _k(t)\) of the Eq. (1.1) are interlacing:$$\begin{aligned} \nu _0(t)>\mu _1>\nu _1(t)>\mu _2>\nu _2(t)> \cdots>\nu _{n-1}(t)>\mu _n>\nu _{n}(t), \quad \forall \,t\in \mathbb {R}.\nonumber \\ \end{aligned}$$ (1.10)In particular for \(t=0\), the roots \(\nu _k(0)=\lambda _k\) of the Eq. (1.1) are the roots of the polynomial P:$$\begin{aligned}&P(z)=(z-\lambda _0)\cdot (z-\lambda _1)\cdot \,\,\cdots \,\,\cdot (z-\lambda _n),\end{aligned}$$ (1.11)$$\begin{aligned}&\lambda _0>\mu _1>\lambda _1>\mu _2>\lambda _2>\cdots>\lambda _{n-1}>\mu _n>\lambda _{n}. \end{aligned}$$ (1.12)Since \(R^{\prime }(x)>0\) for \(x\in \mathbb {R},\,x\not =\mu _1,\ldots ,\mu _n\), each of the functions \(\nu _k(t),k=0,1,\ldots ,n\), can be continued as a single valued holomorphic function to some neighborhood of \(\mathbb {R}\). However the functions \(\nu _k(t)\) can not be continued as single-valued analytic functions to the whole complex t-plane. According to (1.4),$$\begin{aligned} R^{\prime }(z)=\frac{P^{\prime }(z)Q(z)-Q^{\prime }(z)P(z)}{Q^2(z)}. \end{aligned}$$ (1.13)The polynomial \(P^{\prime }Q-Q^{\prime }P\) is of degree 2n and is strictly positive on the real axis. Therefore this polynomial has n roots \(\zeta _1,\ldots ,\zeta _n\) in the upper half-plane \(\text {Im}(z)>0\) and n roots \(\overline{\zeta _1},\ldots ,\overline{\zeta _n}\) in the lower half-plane \(\text {Im}(z)<0\). (Not all roots \(\zeta _1,\ldots ,\zeta _n\) must be distinct.) The points \(\zeta _1,\ldots ,\zeta _n\) and \(\overline{\zeta _1},\ldots ,\overline{\zeta _n}\) are the critical points of the function R: \(R^{\prime }(\zeta _k)=0,\,R^{\prime }(\overline{\zeta _k})=0,\ 1\le k\le n.\) The critical values \(t_k=R(\zeta _k),\,\overline{t_k}=R(\overline{\zeta _k}),\ 1\le k\le n,\) of the function R are the ramification points of the function \(\nu (t)\):$$\begin{aligned} R(\nu (t))=t \end{aligned}$$ (1.14)(Even if the critical points \(\zeta ^{\prime }\) and \(\zeta ^{\prime \prime }\) of R are distinct, the critical values \(R(\zeta ^{\prime })\) and \(R(\zeta ^{\prime \prime })\) may coincide.) We denote the set of critical values of the function R by \(\mathcal {V}\):$$\begin{aligned} \mathcal {V}=\mathcal {V}^{+}\cup \mathcal {V}^{-},\quad \mathcal {V}^{+}=\{t_1,\,\ldots \,,t_n\},\ \mathcal {V}^{-}=\{\overline{t_1},\,\ldots \,,\overline{t_n}\}. \end{aligned}$$ (1.15)Not all values \(t_1,\,\ldots \,,t_n\) must be distinct. However \(\mathcal {V}\not =\emptyset \). In view of (1.9), \(\text {Im}\,t_k>0,\,1\le k\le n\). So$$\begin{aligned} \mathcal {V}^{+}\subset \{t\in \mathbb {C}:\,\text {Im}\,t>0\},\quad \mathcal {V}^{-}\subset \{t\in \mathbb {C}:\,\text {Im}\,t<0\}. \end{aligned}$$ (1.16)Let G be an arbitrary simply connected domain in the t-plane which does not intersect the set \(\mathcal {V}\). Then the roots of Eq. (1.1) are pairwise distinct for each \(t\in {}G\). We can enumerate these roots, say \(\nu _0(t),\nu _1(t),\,\ldots \,\nu _n(t)\), such that all functions \(\nu _k(t)\) are holomorphic in G.The strip \(S_h\),$$\begin{aligned} S_h=\{t\in \mathbb {C}:|\text {Im}\,t|<h\},\ \ \text {where} \ \ h=\min \limits _{1\le k\le n}\!\text {Im}\,t_k, \end{aligned}$$ (1.17)does not intersect the set \(\mathcal {V}\). So \(n+1\) single valued holomorphic branches of the function \(\nu (t)\), (1.14), are defined in the strip \(S_h\). We choose such enumeration of these branches which agrees with the enumeration (1.10) on\(\mathbb {R}\).From (1.6) and (1.2) it follows that the polynomial P is representable in the form $$\begin{aligned} P(z)=z\,Q(z)-\sum \limits _{k=1}^{n}\alpha _kQ_k(z), \end{aligned}$$ (1.18a)where$$\begin{aligned} Q_k(z)=Q(z)/(z-\mu _k),\quad k=1,2,\,\ldots \,,n. \end{aligned}$$ (1.18b)2 Determinant Representation of the Polynomial Pencil \({P(z)-tQ(z)}\)The polynomial pencil \(P(z)-tQ(z)\) is hyperbolic: for each real t, all roots of the Eq. (1.8) are real.