Hyperbolicity with weight of polynomials in terms of comparing their power

For a given completely regular Newton polyhedron $\mathfrak{R}$, and a given vector $N\in\mathbb{R}^n$, we give conditions under which a weakly hyperbolic polynomial (with respect to the vector $N$) $P(\xi)=P(\xi_1,\dots,\xi_n)$ is $\mathfrak{R}$-hyperbolic (with respect to the vector $N$). For polynomials of two variables, the largest number $s >0$ is determined for which an $\mathfrak{R}$-hyperbolic (with respect to the vector $N$) polynomial is $s$-hyperbolic.