Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-16T17:13:40.759Z Has data issue: false hasContentIssue false

ON THE NUMBER OF REAL ZEROS OF POLYNOMIALS OF EVEN DEGREE

Published online by Cambridge University Press:  27 February 2023

LANDE MA*
Affiliation:
School of Mathematical Sciences, Tongji University, Shanghai 200092, PR China
ZHAOKUN MA
Affiliation:
YanZhou College, ShanDong Radio and TV University, YanZhou, ShanDong 272100, PR China e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

For any real polynomial $p(x)$ of even degree k, Shapiro [‘Problems around polynomials: the good, the bad and the ugly$\ldots $’, Arnold Math. J. 1(1) (2015), 91–99] proposed the conjecture that the sum of the number of real zeros of the two polynomials $(k-1)(p{'}(x))^{2}-kp(x)p{"}(x)$ and $p(x)$ is larger than 0. We prove that the conjecture is true except in one case: when the polynomial $p(x)$ has no real zeros, the derivative polynomial $p{'}(x)$ has one real simple zero, that is, $p{'}(x)=C(x)(x-w)$, where $C(x)$ is a polynomial with $C(w)\ne 0$, and the polynomial $(k-1)(C(x))^2(x-w)^{2}-kp(x)C{'}(x)(x-w)-kC(x)p(x)$ has no real zeros.

Type
Research Article
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

1 Introduction

The assertion that if a real polynomial $p(x)$ has only simple real zeros, then the function $p(x)$ is (locally) strictly monotone was known to Gauss (see [Reference Sheil-Small3]). We can reformulate it in the form of the classical Laguerre inequality: if $p(x)$ has only simple real zeros, then the polynomial $p_1(x)=(p{'}(x))^{2}-p(x)p{"}(x)$ is strictly positive. A refinement of the Laguerre inequality constitutes the Hawaiian conjecture (see [Reference Csordas, Craven and Smith1]), where if $p(x)$ is a real polynomial, then the number of real zeros of $({p{'}(x)}/{p(x)}){'}$ does not exceed the number of nonreal zeros of $p(x)$ . The Hawaiian conjecture was settled in 2011 by Tyaglov [Reference Tyaglov4]. Shapiro proposed three conjectures around the Hawaiian conjecture (see Conjectures 11, 12 and 13 in [Reference Shapiro2]). Conjecture 11 is discussed in [Reference Tyaglov and Atia5].

We consider Conjecture 12 which states: for any real polynomial $p(x)$ of even degree k, we have $\Delta := \sharp _r [(k-1)(p{'}(x))^{2}-kp(x)p{"}(x)]+\sharp _r p(x)>0$ . Here, $\sharp _r p(x)$ stands for the number of real zeros of a polynomial $p(x)$ with real coefficients.

Our first result shows that, in most cases, the conjecture is true.

Theorem 1.1. Let $p(x)$ be a real polynomial of even degree k. Then the quantity $\Delta = \sharp _r [(k-1)(p{'}(x))^{2}-kp(x)p{"}(x)]+\sharp _r p(x)>0$ if and only if one of the following four cases holds:

  1. (1) the polynomial $p(x)$ has real zeros;

  2. (2) the polynomial $p(x)$ has no real zeros and the polynomial $p{'}(x)$ has at least three distinct real zeros;

  3. (3) the polynomial $p(x)$ has no real zeros and the polynomial $p{'}(x)$ has one real zero with exponent larger than 1;

  4. (4) the polynomial $p(x)$ has no real zeros, the polynomial $p{'}(x)$ has one real zero which is simple, that is, $p{'}(x)=C(x)(x-w)$ , where $C(x)$ is a polynomial with $C(w)\ne 0$ , and the polynomial $(k-1)(C(x))^2(x-w)^{2}-kp(x)C{'}(x)(x-w)-kC(x)p(x)$ has at least one real zero.

The only case in which the conjecture is false is described in our second result.

Theorem 1.2. Let $p(x)$ be a real polynomial of even degree k. Then the quantity $\Delta = \sharp _r [(k-1)(p{'}(x))^{2}-kp(x)p{"}(x)]+\sharp _r p(x)=0$ if and only if the polynomial $p(x)$ has no real zeros, the polynomial $p{'}(x)$ has one real zero which is simple, that is, $p{'}(x)=C(x)(x-w)$ , where $C(x)$ is a polynomial with $C(w)\ne 0$ , and the polynomial $(k-1)(C(x))^2(x-w)^{2}-kp(x)C{'}(x)(x-w)-kC(x)p(x)$ has no real zeros.

At the end of the paper, we give some examples to show that the case described in Theorem 1.2 does occur.

2 Proofs of the theorems

We derive Theorem 1.1 from a series of lemmas.

Lemma 2.1. For a real polynomial $p(x)$ of even degree k, the real zeros of the polynomial $kp{"}(x)p(x)-(k-1)(p{'}(x))^2$ are all included in the critical points of the rational fraction $P(x)={(p{'}(x))^{k}}/{(p(x))^{k-1}} $ .

Proof. Observe that

$$ \begin{align*} P{'}(x)=\bigg(\frac{(p{'}(x))^{k}}{(p(x))^{k-1}}\bigg)' & =\frac{k(p{'}(x))^{k-1}p{"}(x)(p(x))^{k-1}-(k-1)(p{'}(x))^{k}(p(x))^{k-2}p{'}(x)}{(p(x))^{2k-2}} \\ & =\frac{k(p{'}(x))^{k-1}p{"}(x)(p(x))^{k-1}-(k-1)(p{'}(x))^{k+1}(p(x))^{k-2} }{(p(x))^{2k-2}} \\ & =\frac{(p{'}(x))^{k-1}(kp{"}(x)p(x)-(k-1)(p{'}(x))^2)}{(p(x))^{k}}.\\[-3.3pc] \end{align*} $$

Lemma 2.2. When the real polynomial $p(x)$ of even degree has real zeros, we have $ \sharp _r [(k-1)(p{'}(x))^{2}-kp(x)p{"}(x)]+\sharp _r p(x)>0$ .

Now suppose $p(x)$ is a real polynomial of even degree with no real zeros, so that $\sharp _r p(x)=0$ . The derivative polynomial $p{'}(x)$ has odd degree. A real polynomial of odd degree has an odd number of real zeros. In particular, it has at least one real zero.

Lemma 2.3. Let $p(x)$ be a real polynomial of even degree with no real zeros. If $p{'}(x)$ has at least three distinct real zeros, then $ \sharp _r [(k-1)(p{'}(x))^{2}-kp(x)p{"}(x)]+\sharp _r p(x)>0$ .

Proof. The rational function $P(x)$ is a real function. Since $p(x)$ has no real zeros and $p{'}(x)$ has no real poles, the rational function $P(x)$ has no real poles and so satisfies the conditions of Rolle’s theorem. By the hypothesis, the polynomial $p{'}(x)$ has at least three real zeros. By Rolle’s theorem, between two adjacent real zeros of $P(x)$ , there is at least one real critical point. So, $P(x)$ has at least two real critical points. These two real critical points of $P(x)$ are not zeros of $p{'}(x)$ . So, by Lemma 2.1, at least two real critical points of $P(x)$ are real zeros of the polynomial $(k-1)(p{'}(x))^{2}-kp(x)p{"}(x)$ . So, $ \sharp _r [(k-1)(p{'}(x))^{2}-kp(x)p{"}(x)]\ge 2>0$ .

Example 2.4. Let $p_1(x)=x^4-2x^2+5=(x^2-1)^2+1$ , so $k=4$ .

Obviously, $p_1(x) $ has four distinct complex zeros and it has no real zeros. Further, $p_1^{\prime }(x)=4x^3-4x=4x(x^2-1)$ has three real zeros. In each of the intervals $(-1,0)$ and $(0,1)$ , there is one critical point of the rational fraction $P_1(x)={(p{'}(x))^{k}}/{p^{k-1}(x)}={(4x^3-4x)^4}/{(x^4-2x^2+5)^3}$ and $ \sharp _r [(k-1)(p_1^{\prime }(x))^{2}-kp_1(x)p_1^{\prime \prime }(x)]=2>0$ . This is in accord with Lemma 2.3.

Lemma 2.5. Let $p(x)$ be a real polynomial of even degree with no real zeros. If $p{'}(x)$ has one real zero with exponent larger than 1, then $ \sharp _r [(k-1)(p{'}(x))^{2}-kp(x)p{"}(x)]+\sharp _r p(x)>0$ .

Proof. By hypothesis, $p{'}(x)=C(x)(x-w)^l$ , where $C(x)$ is a polynomial, w is real, $C(w)\ne 0$ and $l>1$ . Then,

$$ \begin{align*} &(k -1)(p{'}(x))^{2}-kp(x)p{"}(x) \\ &\quad=(k-1)(C(x))^2(x-w)^{2l}-kp(x)C{'}(x)(x-w)^l-klC(x)p(x)(x-w)^{l-1} \\ &\quad=(x-w)^{l-1}((k-1)(C(x))^2(x-w)^{l+1}-kp(x)C{'}(x)(x-w)-klC(x)p(x)) \end{align*} $$

and this polynomial has a zero at $z=w$ with exponent $l-1$ . From this, it follows that $ \sharp _r [(k-1)(p{'}(x))^{2}-kp(x)p{"}(x)]+\sharp _r p(x)\ge l-1>0$ .

Lemma 2.6. Let $p(x)$ be a real polynomial of even degree with no real zeros. If $p{'}(x)$ has one real zero which is simple, that is, $p{'}(x)=C(x)(x-w)$ , where $C(x)$ is a polynomial with $C(w)\ne 0$ , and $(k-1)(C(x))^2(x-w)^{2}-kp(x)C{'}(x)(x-w)-kC(x)p(x)$ has real zeros, then $ \sharp _r [(k-1)(p{'}(x))^{2}-kp(x)p{"}(x)]+\sharp _r p(x)>0$ .

Proof. By hypothesis, the polynomial

$$ \begin{align*}(k-1)(p{'}(x))^{2}-kp(x)p{"}(x)=(k-1)(C(x))^2(x-w)^{2}-kp(x)C{'}(x)(x-w)-kC(x)p(x)\end{align*} $$

has real zeros. Consequently, $ \sharp _r [(k-1)(p{'}(x))^{2}-kp(x)p{"}(x)]+\sharp _r p(x)>0$ .

Proof of Theorem 1.1

Let $\Delta = \sharp _r [(k-1)(p{'}(x))^{2}-kp(x)p{"}(x)]+\sharp _r p(x)$ . The four cases of Theorem 1.1 arise as follows.

  1. (1) If $p(x)$ has real zeros, then $\Delta>0$ by Lemma 2.2.

  2. (2) If $p(x)$ has no real zeros and $p{'}(x)$ has at least three distinct real zeros, then $\Delta>0$ by Lemma 2.3.

  3. (3) Suppose $p{'}(x)$ has fewer than three distinct real zeros. Because $p'(x)$ is a polynomial of odd degree, it must have just one real zero. If $p(x)$ has no real zeros and $p{'}(x)$ has one real zero with exponent larger than 1, then $\Delta>0$ by Lemma 2.5.

  4. (4) If $p(x)$ has no real zeros, $p{'}(x)=C(x)(x-w)$ has one real zero which is simple, and the polynomial $(k-1)(C(x))^2 (x-w)^{2}-kp(x)C{'}(x)(x-w)-kC(x)p(x)$ has real zeros, then $\Delta>0$ by Lemma 2.6.

The only remaining case is when $p(x)$ has no real zeros, $p{'}(x)=C(x)(x-w)$ has one real zero which is simple, that is, $C(x)$ is a polynomial with $C(w)\ne 0$ , and the polynomial $(k-1)(C(x))^2(x-w)^{2}-kp(x)C{'}(x)(x-w)-kC(x)p(x)$ has no real zeros. In this case, the calculation in Lemma 2.6 shows that $\Delta = 0$ . This completes the proof of Theorem 1.1.

Proof of Theorem 1.2

Let $\Delta = \sharp _r [(k-1)(p{'}(x))^{2}-kp(x)p{"}(x)]+\sharp _r p(x)$ . From the proof of Theorem 1.1, the hypotheses of Theorem 1.2 describe the only case in which $\Delta =0$ ; in all other cases, $\Delta>0$ .

Example 2.7. Let $p_2(x)=x^2+ax+b$ with $a,b$ real, so $k=2$ .

For this example, $(k-1)(p_2^{\prime }(x))^{2}-kp_2(x)p{"}(x)= (2x+a)^2-4(x^2+ax+b)= a^2-4b$ . If $a^2-4b<0$ , then the polynomials $(k-1)(p_2^{\prime }(x))^{2}-kp_2(x)p_2^{\prime \prime }(x)$ and $p_2(x)$ have no real zeros, that is, $ \sharp _r [(k-1)(p_2^{\prime }(x))^{2}-kp_2(x)p_2^{\prime \prime }(x)]+\sharp _r p_2(x)=0$ , in contrast to Shapiro’s conjecture.

Example 2.8. Let $p_3(x)=x^4+x^2+1$ , so $k=4$ . For this example, $(k-1)(p_3^{\prime }(x))^{2}- kp_3(x)p_3^{\prime \prime } (x){\kern-1pt}={\kern-1pt}3(4x^3{\kern-1pt}+{\kern-1pt}2x)^2{\kern-1pt}-4(x^4{\kern-1pt}+{\kern-1pt}x^2{\kern-1pt}+{\kern-1pt}1)(12x^2{\kern-1pt}+{\kern-1pt}2)=-4(2x^4{\kern-1pt}+{\kern-1pt}11x^2{\kern-1pt}+{\kern-1pt}2)$ . The zeros of the polynomial $2t^2+11t+2$ are $\tfrac 12(-11\pm \sqrt {105})$ which are both negative real zeros. So, the polynomial $2x^4+11x^2+2$ has four complex zeros and no real zeros. So, $ \sharp _r [(k-1)(p_3^{\prime }(x))^{2}-kp_3(x)p_3^{\prime \prime }(x)]+\sharp _r p_3(x)=0$ .

References

Csordas, G., Craven, T. and Smith, W., ‘The zeros of derivatives of entire functions and the Wiman–Pólya conjecture’, Ann. of Math. (2) 125(2) (1987), 405431.Google Scholar
Shapiro, B., ‘Problems around polynomials: the good, the bad and the ugly $\dots$ ’, Arnold Math. J. 1(1) (2015), 9199.CrossRefGoogle Scholar
Sheil-Small, T., Complex Polynomials, Cambridge Studies in Advanced Mathematics, 75 (Cambridge University Press, Cambridge, 2002).CrossRefGoogle Scholar
Tyaglov, M., ‘On the number of critical points of logarithmic derivatives and the Hawaii conjecture’, J. Anal. Math. 114 (2011), 162.CrossRefGoogle Scholar
Tyaglov, M. and Atia, M. J., ‘On the number of non-real zeroes of a homogeneous differential polynomial and a generalisation of the Laguerre inequalities’, J. Math. Anal. Appl. 494(2) (2021), Article no. 124652.CrossRefGoogle Scholar