Let $k$ be a field, let $n,d$ be nonnegative integers, and let $F \in k[X_0,\ldots,X_n]$ be homogeneous of degree $d$. If $(a_0,\ldots,a_n),(b_0,\ldots,b_n) \in k^{n+1}\setminus\{0\}$ represent the same point of $\mathbb{P}^n_k$, so that
Consequently, for $P=[a_0:\cdots:a_n]\in \mathbb{P}^n_k$, the condition $F(a_0,\ldots,a_n)=0$ is independent of the chosen [homogeneous coordinates](/page/Homogeneous%20Coordinates) for $P$.