A map $f:X\to Y$ between quasi-affine varieties $X\subset k^n,Y\subset k^m$ over a field $k$ is called algebraic if there is a map $f':k^n\to k^m$ whose component functions are polynomials, such that $f'$ restricts to $f$ on $X$ .

Alternatively, $f$ is algebraic if the pullback map $f^*:C(Y)\to C(X)$ maps the coordinate ring of $Y$ , $k[Y]$ , to the coordinate ring of $X$ , $k[X]$ .