4・76 Let $n \geqslant 4, \alpha_{1}, \alpha_{2}, \cdots, \alpha_{n} ; \beta_{1}, \beta_{2}, \cdots, \beta_{n}$ be two sets of real numbers, satisfying
$$
\sum_{j=1}^{n} \alpha_{j}^{2}<1, \sum_{j=1}^{n} \beta_{j}^{2}<1 .
$$

Let
$$
\begin{array}{l}
A^{2}=1-\sum_{j=1}^{n} \alpha_{j}^{2}, B^{2}=1-\sum_{j=1}^{n} \beta_{j}^{2}, \\
W=\frac{1}{2}\left(1-\sum_{j=1}^{n} \alpha_{j} \beta_{j}\right)^{2} .
\end{array}
$$

Find all real numbers $\lambda$, such that the equation
$$
x^{n}+\lambda\left(x^{n-1}+\cdots+x^{3}+w x^{2}+A B x+1\right)=0
$$

has only real roots.
［Solution］ Obviously, when $\lambda=0$, the equation in the problem has only real roots.
Below, we prove that when $\lambda \neq 0$, the roots of the equation in the problem cannot all be real.
In fact, if $\lambda \neq 0$, and the $n$ roots of the equation in the problem are all real. At this time, we denote these $n$ roots as
$$
r_{1}, r_{2}, \cdots, r_{n} \text {. }
$$

Clearly, these $n$ roots are not equal to 0. Furthermore, by the relationship between roots and coefficients, we have
$$
\left\{\begin{array}{l}
r_{1} r_{2} \cdots \cdots r_{n}=(-1)^{n} \lambda, \\
r_{1} r_{2} \cdots \cdots r_{n}\left(\frac{1}{r_{1}}+\frac{1}{r_{2}}+\cdots+\frac{1}{r_{n}}\right)=(-1)^{n-1} \lambda A B, \\
r_{1} r_{2} \cdots \cdots r_{n} \cdot\left(\sum_{1 \leqslant j0 .
\end{aligned}
\right.
$$

On the other hand, by the given conditions,
$$
\begin{aligned}
A^{2} B^{2} & =\left(1-\sum_{j=1}^{n} \alpha_{j}^{2}\right)\left(1-\sum_{j=1}^{n} \beta_{j}^{2}\right) \\
& \leqslant\left[\frac{\left(1-\sum_{j=1}^{n} \alpha_{j}^{2}\right)+\left(1-\sum_{j=1}^{n} \beta_{j}^{2}\right)}{2}\right]^{2} \\
& =\left(1-\frac{1}{2} \sum_{j=1}^{n}\left(\alpha_{j}^{2}+\beta_{j}^{2}\right)\right)^{2} \\
& \leqslant\left(1-\sum_{j=1}^{n} \alpha_{j} \beta_{j}\right)^{2} \\
& =2 W .
\end{aligned}
$$

Thus, $A^{2} B^{2}-2 W \leqslant 0$, which is a contradiction.
Therefore, when $\lambda \neq 0$, the roots of the equation in the problem cannot all be real.
In summary, the equation in the problem has only real roots if and only if the real number $\lambda=0$.