Answer by F Zaldivar for Understanding analytic irreducibility of homogenous polynomial

This is a basic concept: $h_F$ is the homogenization of $F$ and for the corresponding projective plane curve $X=V(h_F)\subseteq{\mathbb P}^2$, for the local ring of $X$ at the point $P=[0,1,1]$, the definition asks that its completion has no zero divisors.