  A mixed polynomial $f(\boldsymbol{z}, \bar{\boldsymbol{z}})$ is called a
mixed weighted homogeneous polynomial (Definition 5) if it is both radially and
polar weighted homogeneous. Let $f$ be a mixed weighted homogeneous polynomial
with respect to a strictly positive radial weight vector $P$ and a polar weight
vector $Q$. Suppose that $f$ is Newton non-degenerate over a compact face
$\Delta(P)$ and polar weighted homogeneous of non-zero polar degree with
respect to $Q$. Then $f : {{\mathbb{C}}^*}^n \to \mathbb{C}$ has no mixed
critical points. Moreover, under the assumption $f^{-1}(0) \cap
{{\mathbb{C}}^*}^n \neq \emptyset$, $f : {{\mathbb{C}}^*}^n \to \mathbb{C}$ is
surjective. In other words, in this situation, Newton non-degeneracy over a
compact face $\Delta(P)$ implies strong Newton non-degeneracy over $\Delta(P)$
(Proposition 10). With this fact as a starting point, we investigate the sets
$f^{-1}(0) \cap {{\mathbb{C}}^*}^n$, and show the existence of a collection of
mixed weighted homogeneous polynomials $f = f_{\Delta (P)}$ of non-zero polar
degree which satisfy $\dim \Delta (P) \geq 1$ and $f^{-1}(0) \cap
{{\mathbb{C}}^*}^n = \emptyset$ (Theorem 11). We also give an example of
convenient mixed function germs of mixed weighted homogeneous face type which
are not true non-degenerate (Definition 14).
