Definition 22.1.67433FF564E9A3

Definition 22.1.67.433FF564E9A3 A (real-valued) valuation on $X$ (or a valuation of $\mathbb {C}(X)$) is a map $v\colon \mathbb {C}(X)\rightarrow (-\infty ,\infty ]$ satisfying the following conditions:

1. 8EDEF9CF138B

   For $f\in \mathbb {C}(X)$, $v(f)=\infty $ if and only if $f=0$;

2. 3ACA653FD4D8

   For $f,g\in \mathbb {C}(X)$, $v(fg)=v(f)+v(g)$;

3. 4C1AC87714F3

   For $f,g\in \mathbb {C}(X)$, $v(f+g)\geq v(f)\land v(g)$;

4. C9D9521A044A

   For $c\in \mathbb {C}^{\times }$, $v(c)=0$.