Before revision:  Second, we establish that setting \sigma(x)=(x_1^2 + \cdots + x_n^2)^\nu for some integer \nu suffices if P (x) is even, homogeneous , and positive definite  .
After revision:  Second, we establish that setting \sigma(x)=(x_1^2 + \cdots + x_n^2)^\nu for some integer \nu suffices if P (x) is even, homogeneous , and positive definite globally .