ON THE NUMBER OF REAL ZEROS OF POLYNOMIALS OF EVEN DEGREE

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.

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 [3]). 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 [1]), 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 [4]. Shapiro proposed three conjectures around the Hawaiian conjecture (see Conjectures 11, 12 and 13 in [2]). Conjecture 11 is discussed in [5].

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$ .