# 绝对赋值

1. |x| ≥ 0,
2. |x| = 0 当且仅当 x = 0,
3. |xy| = |x||y|,
4. |x + y| ≤ |x| + |y|.

| 1+1+...(n次) | = | −1−1...(n次) | ≤ n.

## 绝对赋值的类型

q = pn(a/b), 其中a，b是不被p整除的整数。
${\displaystyle \left|p^{n}{\frac {a}{b}}\right|_{p}=p^{-n}.}$

## 几何概念联系

${\displaystyle \scriptstyle {\mathfrak {R}}=\mathbb {C} [x,y]}$ 是在复域的两个变量的多项式环${\displaystyle \scriptstyle \mathbb {K} =\mathbb {C} (x,y)}$有理函数，并考虑收敛

${\displaystyle f(x,y)=y-\sum _{n=3}^{\infty }{\frac {x^{n}}{n!}}\in \mathbb {C} \{x,y\}}$

${\displaystyle t}$ 参数化后解析零点集为${\displaystyle \scriptstyle V_{f}\,}$，则作为多项式环形式幂级数环

${\displaystyle V_{f}=\{(x,y)\in \mathbb {C} ^{2}\,|\,f(x,y)=0\}=\left\{(x,y)\in \mathbb {C} ^{2}\,|\,(x,y)=\left(t,\sum _{n=3}^{\infty }t^{n}\right)\right\}}$

${\displaystyle v(P)=\mathrm {ord} _{t}\left(P|_{V_{f}}\right)={\mathrm {ord} }_{t}\left(P\left(t,\sum _{n=3}^{+\infty }t^{n}\right)\right)\quad \forall P\in \mathbb {C} [x,y]}$

${\displaystyle v(P/Q)={\begin{cases}v(P)-v(Q)&\forall P/Q\in {\mathbb {C} (x,y)}^{*}\\\infty &P\equiv 0\in \mathbb {C} (x,y)\end{cases}}}$

${\displaystyle {\begin{array}{l}v(x)=\mathrm {ord} _{t}(t)=1\\v(x^{6}-y^{2})=\mathrm {ord} _{t}(t^{6}-t^{6}-2t^{7}-3t^{8}-\cdots )=\mathrm {ord} _{t}(-2t^{7}-3t^{8}-\cdots )=7\\v\left({\frac {x^{6}-y^{2}}{x}}\right)=\mathrm {ord} _{t}(-2t^{7}-3t^{8}-\cdots )-\mathrm {ord} _{t}(t)=7-1=6\end{array}}}$

## 参考

• Jacobson, Nathan, Valuations: paragraph 6 of chapter 9, Basic algebra II 2nd, New York: W. H. Freeman and Company, 1989 [1980], ISBN 0-7167-1933-9, Zbl 0694.16001. A masterpiece on algebra written by one of the leading contributors.
• Chapter VI of Zariski, Oscar; Samuel, Pierre, Commutative algebra, Volume II, Graduate Texts in Mathematics 29, New York, Heidelberg: Springer-Verlag, 1976 [1960], ISBN 978-0-387-90171-8