Let $k$ be a field, let $n\geq 0$, and let $X\subset \mathbb P_k^n$ be an integral nonsingular projective $k$-variety with $\dim X\geq 1$. Let

\begin{align*} S:=k[X_0,\dots,X_n]. \end{align*}

Let $d\in\mathbb N_0$, and let $F,G\in S_d$ be homogeneous forms of degree $d$ whose restrictions to $X$ are nonzero global sections of $\mathcal O_X(d)$. For a homogeneous form $H\in S_d$ whose restriction to $X$ is nonzero, let $(H)_0|_X$ denote the effective Cartier, equivalently Weil, divisor of zeros of the section $H|_X\in H^0(X,\mathcal O_X(d))$. Then the degree-zero homogeneous quotient $F/G$ defines a nonzero rational function on $X$, and

\begin{align*} \operatorname{div}(F/G)=(F)_0|_X-(G)_0|_X. \end{align*}

In particular,

\begin{align*} (F)_0|_X\sim (G)_0|_X. \end{align*}