Remark on “When are all the zeros of a polynomial real and distinct?”

Abstract

The purpose of this note is to point out that Chamberland’s theorem is implicitly contained in the elementary lore of the theory of orthogonal polynomials.

In [1], Chamberland proved the following theorem:

Let P be a polynomial of degree $n\ge 2$ with real coefficients. Then the zeros of P are real and distinct if and only if

$$\begin{aligned} \Big (\big (P^{(n-j-1)}(x)\big )'\Big )^2-P^{(n-j-1)}(x)\big (P^{(n-j-1)}(x)\big )''>0 \end{aligned}$$

(1)

for all $x\in {\mathbb {R}}$ and $j\in \{1,\dots , n-1\}$.

The purpose of this note is to point out that this result is implicitly contained in the elementary lore of the theory of orthogonal polynomials on the real line (OPRL).

$(\Rightarrow )$ By Geronimus–Wendroff’s theorem [2,  Exercise 5.5, p. 30],¹ there are probability measures on ${\mathbb {R}}$, $\mu _j$, with finite moments so that $P_{j+1}=P^{(n-j-1)}$ and $P_j=P'_{j+1}$ are among the OPRL for $\mu _j$. Since $P_{j+1}$ and $P_j$ have leading coefficients of the same sign, then (see [2,  (4.13), p. 24])

$$\begin{aligned} P'_{j+1}(x)P_j(x)-P'_j(x)P_{j+1}(x)>0 \end{aligned}$$

(2)

for all $x\in {\mathbb {R}}$ and $j\in \{1,\dots , n-1\}$.

$(\Leftarrow )$ Set $P_{j}=P^{(n-j)}$ for all $j\in \{1,\dots , n\}$. Without loss of generality we can assume P monic. Suppose that the zeros of $P_j$ are real and distinct. From (2) we see that $P_{j+1}$ has at least one zero between two zeros of $P_{j}$, and so $P_{j+1}$ has at least $j-1$ real and distinct zeros. Suppose that the other two zeros of $P_{j+1}$ are not real, and therefore they appear as a complex conjugate pair. Let a be the largest zero of $P_j$. Clearly, $P_{j+1}(x)>0$ for all $x>a$. However, by (2), we have $P_{j+1}(a)<0$, which leads to a contradiction. From this we conclude that if $P_{j}$ has real and distinct zeros, then the same holds for $P_{j+1}$. Since $P_1=P^{(n-1)}$ has a single real zero, we infer successively that $P_2, \dots , P_n=P$ have real and distinct zeros.

We emphasize for the reader’s convenience that any polynomial with real and distinct zeros is an element of a sequence of OPRL, and so any sentence starting with “the zeros of a polynomial are real and distinct if and only if” is virtually talking about an element of a certain sequence of OPRL.