regular map

A regular map $\varphi :k^n\to k^m$ between affine spaces over an algebraically closed field is merely one given by polynomials. That is, there are $m$ polynomials $F_1,\mathrm{\dots },F_m$ in $n$ variables such that the map is given by $\varphi (x_1,\mathrm{\dots },x_n)=(F_1(x),\mathrm{\dots },F_m(x))$ where $x$ stands for the many components $x_i$.

A regular map $\varphi :V\to W$ between affine varieties is one which is the restriction of a regular map between affine spaces. That is, if $V\subset k^n$ and $W\subset k^m$, then there is a regular map $\psi :k^n\to k^m$ with $\psi (V)\subset W$ and $\varphi ={\psi |}_V$. So, this is a map given by polynomials, whose image lies in the intended target.

A regular map between algebraic varieties is a locally regular map. That is $\varphi :V\to W$ is regular if around each point $x$ there is an affine variety $V_x$ and around each point $f(x)\in W$ there is an affine variety $W_{f(x)}$ with $\varphi (V_x)\subset W_{f(x)}$ and such that the restriction $V_x\to W_{f(x)}$ is a regular map of affine varieties.