Distribution of Zeros and Critical Points of a Polynomial, and Sendov’s Conjecture

Abstract

According to the Gauss–Lucas theorem, the critical points of a complex polynomial $p(z):=\sum_{j=0}^{n}a_{j}z^{j}$ where $a_{j}\in\mathbb{C}$ always lie in the convex hull of its zeros. In this paper, we prove certain relations between the distribution of zeros of a polynomial and its critical points. Using these relations, we prove the well-known Sendov’s conjecture for certain special cases.

1 INTRODUCTION AND STATEMENT OF RESULTS

Polynomials are the mathematical expressions of the form $p(z):=\sum_{j=0}^{n}a_{j}z^{j}$ where $a_{j}\in\mathbb{C}$ and have been studied since ancient times with regard to their zeros, the values of the variable $z$ that make $p(z)$ vanish. If we plot the zeros of a polynomial $p(z)$ and the zeros of the derived polynomial $p^{\prime}(z)$ (the critical points of $p(z)$) in the complex plane, there are interesting geometric relations between the two sets of points. It was shown by Gauss that the zeros of $p^{\prime}(z)$ are the positions of equilibrium in the force field due to particles of equal mass/charge situated at each zero of $p(z)$, if each particle attracts/repels the other particle with a force equal to the inverse of the distance between them. Concerning the location of critical points of a polynomial , the Gauss–Lucas theorem states that the critical points of a polynomial $p$ lie in the convex hull of its zeros. Regarding the distribution of critical points of $p$ within the convex hull of its zeros the well known Sendov’s Conjecture asserts:

‘‘If all the zeros of a polynomial $p$ lie in $|z|\leq 1$ and if $z_{0}$ is any zero of $p(z)$, then there is a critical point of $p$ in the disk $|z-z_{0}|\leq 1$.’’

The conjecture was posed by Bulgarian mathematician Blagovest Sendov in 1958, but is often attributed to Ilieff because of a reference in Hayman’s Research Problems in Function Theory [5] in 1967. A good number of papers have been published on this conjecture (for details see [6]) but the general conjecture remains open. Rubenstein [11] in 1968 proved the conjecture for all polynomials of degree 3 and 4. In 1969 Schmeisser [12] showed that, if the convex hull containing all zeros of $p$ has its vertices on $|z|=1$, then $p$ satisfies the conjecture (for the proof see [10, Theorem 7.3.4]). Later Schmeisser [13] also proved the conjecture for the Cauchy class of polynomials. In 1996 Borcea [3] showed that the conjecture holds true for polynomials with atmost six distinct zeros and in 1999 Brown and Xiang [4] proved the conjecture for polynomials of degree upto eight. Dégot [6] proved that for every zero (say) $z_{0}$ of a polynomial $p$ there exists lower bound $N_{0}$ depending upon the modulus of $z_{0}$ such that $|z-z_{0}|\leq 1$ contains a critical point of $p$ if deg$(p)>N_{0}$. Chalebgwa [5] gave an explicit formula for such a $N_{0}$. More recent important work in this area includes that of Kumar [8] and Sofi et al. [14]. As for the latest, Terence Tao [15] following on the work of Degot [6], proved that the Sendov’s conjecture holds for polynomials with sufficiently high degree.

In this paper, we prove certain relations between the distribution of zeros of a polynomial $p$ in the complex plane and the distribution of its critical points. Using these relations we prove the Sendov’s Conjecture for the case when all the zeros of a polynomial lie on a circle or a line within the closed unit disk.

Theorem 1.1. Let $p(z):=\sum_{j=0}^{n}a_{j}z^{j}$ be a polynomial with zeros $z_{1},z_{2},\cdots,z_{n}$ . Suppose $z_{1}$ is a zero of $p$ such that $|z_{j}-z_{1}+re^{i\theta}|\leq r$ for all $2\leq j\leq n$ , $0<\theta\leq 2\pi$ and $r>0$, then $|z-z_{1}|\leq r$ always contains a critical point of $p$.

In case $z_{1}$ is the largest zero of $p$ in modulus, then $|z_{j}|\leq|z_{1}|$ for all $2\leq j\leq n,$ and therefore we have $|z_{j}-z_{1}+|z_{1}|e^{i\theta}|\leq|z_{1}|$ where $\theta=\textrm{arg}(z_{1})$ for all $2\leq j\leq n$. Hence by Theorem 1.1, $|z-z_{1}|\leq|z_{1}|$ contains a critical point of $p$. So we have proved the following:

Corollary 1.1. If $p(z):=\sum_{j=0}^{n}a_{j}z^{j}$ is a polynomial with all its zeros $z_{1},z_{2},\cdots,z_{n}$ and suppose $z_{1}$ is the zero of $p$ with largest modulus, that is, $|z_{j}|\leq|z_{1}|$ for all $2\leq j\leq n$, then $|z-z_{1}|\leq|z_{1}|$ always contains a critical point of $p$.

Theorem 1.2. Let $p(z):=\sum_{j=0}^{n}a_{j}z^{j}$ be a nonconstant polynomial with all its zeros $z_{1},z_{2},\cdots,z_{n}$ lying inside the closed disk $|z|\leq r$. Suppose all $z_{j},1\leq j\leq n$ lie on a circle or on a line within the closed disk $|z|\leq r$ , then for every $z_{i},1\leq i\leq n$, there exists a critical point of $p$ in $|z-z_{i}|\leq r$.

Taking $r=1,$ Theorem 1.2 shows that the Sendov’s Conjecture holds when all the zeros of $p$ lie on a circle or a line within the disk containing all the zeros of $p$.

Sendov’s conjecture is about how far away are critical points of a polynomial $p$ from a given zero of $p$. In the converse direction we prove the following:

Theorem 1.3. Let $p(z):=\sum_{j=0}^{n}a_{j}z^{j}$ be a nonconstant polynomial with its zeros $z_{1},z_{2},\cdots,z_{n}$ and critical points $\zeta_{1},\zeta_{2},\cdots,\zeta_{n-1}$ .Then for every critical point $\zeta_{j},1\leq j\leq n-1$ there exists a zero $z_{i},1\leq i\leq n$ such that

$$|z_{i}-\zeta_{j}|^{2}\leq|z_{i}|^{2}-|\zeta_{j}|^{2}.$$

(1.1)

and

$$\left|\frac{z_{i}}{2}-\zeta_{j}\right|\leq\left|\frac{z_{i}}{2}\right|.$$

(1.2)

Remark 1. In the above result if we assume that all the zeros of $p$ lie in the closed unit disk, we get from Theorem 1.3 that for every critical point $\zeta_{j},1\leq j\leq n-1$ of $p$ there exists a zero $z_{i},1\leq i\leq n$ of $p$ such that

$$|z_{i}-\zeta_{j}|\leq 1$$

(1.3)

and $\left|\frac{z_{i}}{2}-\zeta_{j}\right|\leq\frac{1}{2}$, which is the main result by Aziz in [1]. Our proof of the result from which it follows is very short and simple.

Remark 2. If a unique zero of $p$ satisfies (1.3) for all $\zeta_{j},1\leq j\leq n-1$ then by Biernacki’s theorem [10, Theorem 4.5.2] Sendov’s conjecture is true for $p$. Therefore, if Sendov’s conjecture is false, (1.3) must be true for at least two different zeros of $p$ and hence for any polynomial $p$ with all its zeros in the closed unit disk there will always be at least two different zeros say $z_{1}$, $z_{2}$ such that $|z-z_{i}|\leq 1$ for $i=1,2$ contains some critical point of $p$.

2 PROOFS OF THE THEOREMS

Proof of Theorem 1.1. Let $\zeta_{1},\zeta_{2}\dots,\zeta_{n-1}$ be the critical points of $p$ and assume to the contrary $|z_{1}-\zeta_{j}|>r$ for all $1\leq j\leq n-1$.Then

$$\left|\frac{p^{\prime\prime}(z_{1})}{p^{\prime}(z_{1})}\right|=\left|\sum_{j=1}^{n-1}\frac{1}{z_{1}-\zeta_{j}}\right|<\frac{n-1}{r}.$$

Let $p(z)=a_{n}(z-z_{1})q(z),$ where $q(z)=\prod_{j=2}^{n}(z-z_{j})$. Then $p^{\prime}(z_{1})=q(z_{1})$ and $p^{\prime\prime}(z_{1})=2q^{\prime}(z_{1}),$ so that

$$\left|\frac{2q^{\prime}(z_{1})}{q(z_{1})}\right|=\left|\frac{p^{\prime\prime}(z_{1})}{p^{\prime}(z_{1})}\right|<\frac{n-1}{r}$$

and hence

$$\left|\frac{q^{\prime}(z_{1})}{q(z_{1})}\right|=\left|\sum_{j=2}^{n}\frac{1}{z_{1}-z_{j}}\right|<\frac{n-1}{2r}.$$

((*))

Now, by our assumption, for all $2\leq j\leq n$

$$\left|\frac{z_{j}-z_{1}+re^{i\theta}}{re^{i\theta}}\right|\leq 1,$$

we have

$$\mathfrak{Re}\left(\dfrac{1}{1-\dfrac{z_{j}-z_{1}+re^{i\theta}}{re^{i\theta}}}\right)\geq\dfrac{1}{2}.$$

Equivalently, $\mathfrak{Re}\left(\dfrac{re^{i\theta}}{z_{1}-z_{j}}\right)\geq\dfrac{1}{2}$. This gives

$$\mathfrak{Re}\left(\sum_{j=2}^{n}\dfrac{re^{i\theta}}{z_{1}-z_{j}}\right)\geq\dfrac{n-1}{2}.$$

Therefore

$$\left|\sum_{j=2}^{n}\dfrac{re^{i\theta}}{z_{1}-z_{j}}\right|\geq\dfrac{n-1}{2}.$$

Hence,

$$\left|\frac{q^{\prime}(z_{1})}{q(z_{1})}\right|=\left|\sum_{j=2}^{n}\frac{1}{z_{1}-z_{j}}\right|\geq\frac{n-1}{2r},$$

a contradiction to (*). $\Box$

Proof of Theorem 1.2.

Case I : All the zeros of $p$ lie on the circle.

Since the relative distances between zeros and critical points will not change under a translation we may assume without loss of generality that all the zeros of $p$ lie on a circle with centre at the origin and in that case $|z_{i}|=|z_{j}|$ for all $1\leq i,j\leq n$. Therefore by Corollary 1.1 for every $i,1\leq i\leq n$, there exists a zero of $p^{\prime}$ inside $|z-z_{i}|\leq|z_{j}|\leq r$.

Case II : All the zeros of $p$ lie on a line.

Again since a rigid motion will not alter the relative distances between zeros and critical points of $p$, it is sufficient to prove the result in case all the zeors of $p$ lie on the real line. Let $\zeta_{1},\zeta_{2}\dots,\zeta_{n-1}$ be the critical points of $p$ and assume to the contrary that there exists a zero of $p$ say $z_{1}$ such that $|z_{1}-\zeta_{j}|>r$ for all $1\leq j\leq n-1$. By the same argument as in the proof of Theorem 1.1 we have

$$\left|\sum_{j=2}^{n}\frac{1}{z_{1}-z_{j}}\right|<\frac{n-1}{2r}.$$

(2.1)

We may also assume that $z_{1}>0$. If there exists any zero of $p$, say $z_{j}$ such that $z_{j}>z_{1}$, then by Rolle’s theorem $p$ will have a critical point between $z_{1}$ and $z_{j}$ which will be within a distance $r$ of $z_{1}$ and there is nothing to prove (see Fig. 1). So we may assume $z_{j}\leq z_{1}$ for all $2\leq j\leq n$ and hence $0\leq z_{1}-z_{j}\leq 2r$ for all $2\leq j\leq n$. This implies that for all $2\leq j\leq n$ $\frac{1}{z_{1}-z_{j}}\geq\frac{1}{2r}$, which gives

$$\left|\sum_{j=2}^{n}\frac{1}{z_{1}-z_{j}}\right|=\sum_{j=2}^{n}\frac{1}{z_{1}-z_{j}}\geq\frac{n-1}{2r}$$

a contradiction to (2.1) and hence the result. $\Box$

Fig. 1

If $z_{j}>z_{1}$ for some $2\leq j\leq n$ then $|z-z_{1}|\leq r$ contains a critical point of $p$.

Proof of Theorem 1.3. To prove (1.1) assume the contrary. Therefore, there exist a critical point say $\zeta_{1}$ of $p$ such that

$$|z_{i}-\zeta_{1}|^{2}>|z_{i}|^{2}-|\zeta_{1}|^{2},\quad 1\leq i\leq n.$$

(2.2)

Without loss of generality, we may assume that $\zeta_{1}\geq 0.$ From (2.3) we get

$$|z_{i}|^{2}+\zeta_{1}^{2}-2\zeta_{1}{Re}(z_{i})>|z_{i}|^{2}-\zeta_{1}^{2},\quad 1\leq i\leq n.$$

This gives ${Re}(z_{i})<\zeta_{1}\text{for all}\;1\leq i\leq n$. But this implies that $\zeta_{1}$, a critical point of $p$, lies outside the convex hull of $z_{1},z_{2},\cdots,z_{n}$, the zeros of $p$ violating the Gauss–Lucas theorem and this contradiction proves (1.1). A similar argument proves (1.2). $\Box$