Using Polya's Theorem to check positivity of a multivariate polynomial

Question

I wish to check if a homogeneous polynomial of total degree 4 is positive definite. The polynomial is of the form $$P(u,v,x,y) = \sum_i\alpha_iu^{i_1}v^{i_2}x^{i_3}y^{i_4}$$ with $0 \le i_j \le 2$, and $\vec u = (u,v) \in \mathbb R^2, \vec x = (x,y) \in \mathbb R^2$. By Polya's Theorem, if $P$ is positive then we know there exists $N$ such that $$(u+v+x+y)^N P(u,v,x,y) \ge 0$$ if $u+v+x+y = 1$ on the positive simplex in $\mathbb R^4$. However is it also true that there exists $N_1, N_2$ such that $$(u+v)^{N_1}(x+y)^{N_2}P(u,v,x,y) \ge0$$ for $u+v = 1$ and $x+y = 1$? Thanks for any help or advice.