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Rolle's Theorems for Complex Polynomials

  • Oct. 27, 2011
  • 3:30 p.m.
  • LeConte 412

Abstract

The classical Rolle's theorem states that if $p(x)$ is a real polynomial and $p(-1)=p(1)$, then there exists $\xi \in (-1,1)$, such that $p'(\xi) = 0$. This interval can not be shortened, if the polynomials are of arbitrary degree. But for sets of polynomials of finite dimension, there exists smaller intervals with this property. One such example is, see [1, pp. 203]:

Theorem 1 (Lagguerre-Cesàro) If $p(x)$ is a polynomial of degree $n \geq 2$ with only real zeros and $a=-1, b=1$ are two consecutive zeros of $p(x)$, then at least one zero of $p'(x)$ is in the segment $[-1+2/n, 1-2/n]$. The segment $[-1+2/n, 1-2/n]$ is the smallest segment with this property.

The case of all real polynomials of degree $n\geq 2$ with $p(-1)=p(1)$ was solved by Lubomir Tschakaloff [2], a leading Bulgarian mathematician from the first half of the last century.

Theorem 2 (L. Tschakaloff ) Let $\alpha _ m$ be the biggest zero of the Legendre polynomial of degree $m$. If $p(x)$ is a real polynomial of degree $n\leq 2m$ and $p(-1)=p(1)$, then at least one zero of $p'(x)$ is in the open interval $(-\alpha _ m, \alpha _ m)$ for $n>3$ and in the closed interval $[-\alpha _ 2, \alpha _ 2]$ for $n=3$. If $n=2$, the single zero of $p(x)$ is $\alpha _ 1=0$. Moreover, for every $0\leq \beta _ m <\alpha _ m$, there exists a polynomial of degree $n\leq 2m$ without zeros in the closet interval $[-\beta _ m, \beta _ m]$.

The above two theorems are sharp, as they give the interval with smallest length (measure), containing at least one critical point of a polynomial from a given set of polynomials. We investigate the same problem for complex polynomials.

Definition 1 In this paper, domain is a closed, simply connected and measurable point set $\Omega$ on the complex plane $\mathcal C$ with measure $\mu(\Omega)\geq 0$.

A domain $R _ n$ is called Rolle's domain if for every complex polynomial $p(z)$ of degree $n\geq 2$ and $p(-i)=p(i)$ there exists at least one $\zeta \in R _ n$, such that $p'(\zeta)=0$.

A Theorem $X$ is called Rolle's theorem for complex polynomials if it states that a domain $R _ n^X$ is a Rolle's domain.

A Rolle's theorem $X$ is stronger than the Rolle's theorem $Y$, if $\mu(R _ n^X)<\mu(R _ n^Y)$.

A Rolle's theorem $X$ is sharp, if $\mu(R _ n^X)$ is minimal.

There are several Rolle's theorems for complex polynomials. The most famous one is the Grace-Heawood theorem, see [1, p. 126], which may be formulated as:

Theorem 3 ( Grace-Heawood) The disk

$R _ n^{GH}=D\left(0;\cot\frac{\pi}{n}\right)=\left\{ z:\; |z| \leq \cot\frac{\pi}{n}\right\}$ (1)

is a Rolle's domain.

Another complex Rolle's theorem, see [1, Theorem 4.3.4, p. 128], is the following:

Theorem 4 The double disk $R _ n^F=DD(c;r)=D(-c;r)\bigcup D(c;r)$, where

$c=\cot\frac{\pi}{n-1},\;\;\; r=\sin^{-1}\frac{\pi}{n-1};\;\;\; n\geq 3,$

is a Rolle's domain.

We have

$\mu(R _ n^{GH}) \leq\mu(R _ n^F) \;\;\;{\rm and}\;\;\;\lim _ {n\to\infty}\frac{\mu(R _ n^{GH})}{\mu(R _ n^F)}=\frac{1}{2},$

hence, for $n\geq 3$, Theorem 3 is stronger than Theorem 4.

Theorem 3 is sharp only for $n=2,3$. In this paper we prove the following Rolle's theorem which is stronger than Theorem 3 and sharp for polynomials of degree $n=2,3,4$:

Theorem 5 Let $\nu _ n$ be the only positive zero of the polynomial

$\sum _ {k=0}^{[(n-1)/2]}\frac{n-4k-1}{2k+1}{n-1\choose n-2k-1} x^{n-2k-1}.$

Then, the double disk $R _ n^S=DD(c;r)$, where

$c=\frac{1}{2}\left(\cot\frac{\pi}{n}-\nu _ n^2\tan\frac{\pi}{n}\right),\;\;\; r= \frac{1}{2}\left(\cot\frac{\pi}{n}+\nu _ n^2\tan\frac{\pi}{n}\right)$

is a Rolle's domain.

Theorem 5 is asymptotically sharp in the following sense. If $R _ n^{SS}$ is the Rolle's domain with minimal measure, then

$\lim _ {n\to\infty}\frac{\mu(R _ n^{SS})}{\mu(R _ n^S)}=1.$

To prove the Theorem 5, we develop a method with which it is possible to calculate the minimal Rolle's domain for every natural $n\geq 2$.

Key words: zeros and critical points of polynomials, apolarity, complex Rolle's theorem.

Mathematics Subject Classification: 30C10

References
[1] RAHMAN, Q. I. AND G. SCHMEISSER: Analytic Theory of Polynomials, Oxford Univ. Press Inc., New York, 2002.
[2] TCHAKALOFF, L.: Sur la structure des ensemble linéaires définis par une certaine propriété minimale, Acta Math., 63 (1934), 77 - 97.

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