Let $k$ be an [algebraically closed field](/page/Algebraically%20Closed%20Field), let $m\in\mathbb N$, let $X$ be an irreducible affine variety over $k$, and let $Y\subset \mathbb A_k^m$ be an affine algebraic subset with vanishing ideal $I(Y)\trianglelefteq k[y_1,\dots,y_m]$. For an ideal $J\trianglelefteq k[y_1,\dots,y_m]$, write $V(J):=\{a\in\mathbb A_k^m:F(a)=0\text{ for every }F\in J\}$. Rational maps $\Phi:X\dashrightarrow Y$, where rational maps are taken as equivalence classes of regular maps from nonempty open subsets of $X$ to $Y$, are in bijective correspondence with tuples $(\varphi_1,\dots,\varphi_m)\in k(X)^m$ such that, for every polynomial $F\in I(Y)$, one has $F(\varphi_1,\dots,\varphi_m)=0$ in the function field $k(X)$.

Under this correspondence, a rational map represented by a regular map $\Phi_U:U\to Y$ on a nonempty open subset $U\subset X$ is sent to $(y_1\circ \Phi_U,\dots,y_m\circ \Phi_U)\in k(X)^m$. Conversely, a tuple $(\varphi_1,\dots,\varphi_m)$ satisfying the displayed equations determines the rational map represented on any common nonempty open subset $U\subset X$ on which all $\varphi_i$ are represented by regular functions by $p\mapsto (\varphi_1(p),\dots,\varphi_m(p))$.