Polynomials and Powers

Hello,
let P(X_1,...,X_n) be a polynomial with integer
coefficients. Let us suppose that for any (x_1,...,x_n)
in Z^n such that GCD(x_1,...,x_n)=1, there exist two
integers z and m such that
P(x_1,...,x_n)=z^m.
Can we conclude that P has one of the two forms:
(I) (a_1 X_1+...+a_n X_n)^k
(II) (X_1)^k (X_2)^k ... (X_n)^k?

Thank you for your attention.
Maury
.