  We prove several results on homogeneous plurisubharmonic polynomials on
$\mathbb{C}^n$, $n\in\mathbb{Z}_{\geq 2}$. Said results are relevant to the
problem of constructing local bumpings at boundary points of pseudoconvex
domains of finite D'Angelo $1$-type in $\mathbb{C}^{n+1}$.
