# 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 times) | = | −1−1...(n times) | ≤ 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。
$\left|p^n \frac{a}{b}\right|_p = p^{-n}.$

## 几何概念联系

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

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

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

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

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

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

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

## 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