2005 Putnam B5

Let $P(x_1,\dots,x_n)$ denote a polynomial with real coefficients in the

variables $x_1, \dots, x_n$, and suppose that

$$

\left( \frac{\partial^2}{\partial x_1^2} + \cdots + \frac{\partial^2}{\partial

x_n^2}\right) P(x_1, \dots,x_n) = 0 \quad \text{(identically)}

$$

and that

$$

x_1^2 + \cdots + x_n^2 \text{ divides } P(x_1, \dots, x_n).

$$

Show that $P=0$ identically.

设 $P(x_1,\dots,x_n)$ 表示一个多项式，其系数为

变量 $x_1, \dots, x_n$，并假设

$$

\left( \frac{\partial^2}{\partial x_1^2} + \cdots + \frac{\partial^2}{\partial

x_n^2}\right) P(x_1, \dots,x_n) = 0 \quad \text{(相同)}

$$

还有那个

$$

x_1^2 + \cdots + x_n^2 \text{ 除 } P(x_1, \dots, x_n)。

$$

证明 $P=0$ 相同。