${\displaystyle {}\varphi \colon {{\mathbb {A} }_{K}^{n}}\rightarrow {{\mathbb {A} }_{K}^{n}}}$

${\displaystyle {}\varphi (x_{1},\ldots ,x_{n})=M{\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}}+(v_{1},\ldots ,v_{n})\,,}$

wobei ${\displaystyle {}M}$ eine invertierbare Matrix ist, eine affin-lineare Variablentransformation.

${\displaystyle {}M}$

${\displaystyle {}R}$

${\displaystyle R\times M\longrightarrow M,\,(r,v)\longmapsto rv=r\cdot v,}$

festgelegt ist, die folgende Axiome erfüllt (dabei seien ${\displaystyle r,s\in R}$ und ${\displaystyle u,v\in M}$ beliebig):

1. ${\displaystyle {}r(su)=(rs)u}$,
2. ${\displaystyle {}r(u+v)=(ru)+(rv)}$,
3. ${\displaystyle {}(r+s)u=(ru)+(su)}$,
4. ${\displaystyle {}1u=u}$.

${\displaystyle {}F=\sum _{\nu }a_{\nu }T^{\nu }=\sum _{\nu }a_{\nu }T_{1}^{\nu _{1}}\cdots T_{n}^{\nu _{n}}\,.}$

wobei ${\displaystyle {}a_{\nu }\in R}$ ist für alle ${\displaystyle {}\nu =(\nu _{1},\ldots ,\nu _{n})\in \mathbb {N} ^{n}}$.

${\displaystyle V_{+}({\mathfrak {a}})={\left\{P=(x_{0},\ldots ,x_{n})\in {\mathbb {P} }_{K}^{n}\mid F(P)=0{\text{ für }}{\text{alle homogenen }}F\in {\mathfrak {a}}\right\}}\,}$

das projektive Nullstellengebilde zu ${\displaystyle {}{\mathfrak {a}}}$.