regular map

A regular map $\\phi:k^{n}\\to k^{m}$ between affine spaces over an algebraically closed field is merely one given by polynomials. That is, there are $m$ polynomials $F_{1},\\ldots,F_{m}$ in $n$ variables such that the map is given by $\\phi(x_{1},\\ldots,x_{n})=(F_{1}(x),\\ldots,F_{m}(x))$ where $x$ stands for the many components $x_{i}$ .

A regular map $\\phi:V\\to W$ 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 $\\phi=\\psi|_{V}$ . 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 $\\phi:V\\to W$ 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 $\\phi(V_{x})\\subset W_{f(x)}$ and such that the restriction $V_{x}\\to W_{f(x)}$ is a regular map of affine varieties.

单词	RegularMap
释义	regular map A regular map $\\phi:k^{n}\\to k^{m}$ between affine spaces over an algebraically closed field is merely one given by polynomials. That is, there are $m$ polynomials $F_{1},\\ldots,F_{m}$ in $n$ variables such that the map is given by $\\phi(x_{1},\\ldots,x_{n})=(F_{1}(x),\\ldots,F_{m}(x))$ where $x$ stands for the many components $x_{i}$ . A regular map $\\phi:V\\to W$ 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 $\\phi=\\psi\|_{V}$ . 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 $\\phi:V\\to W$ 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 $\\phi(V_{x})\\subset W_{f(x)}$ and such that the restriction $V_{x}\\to W_{f(x)}$ is a regular map of affine varieties.
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