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# 数学代写|交换代数代写Commutative Algebra代考|Singularities

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## 数学代写|交换代数代写Commutative Algebra代考|Singularities

In Section A.8 we have already defined singular points as points $p$ of a variety $X$ where the local ring $\mathcal{O}{X, p}$ is not regular. In particular, “singular” is a local notion, where “local” so far was mainly considered with respect to the Zariski topology. However, since the Zariski topology is so coarse, small neighbourhoods in the Zariski topology might not be local enough. If our field $K$ is $\mathbb{C}$, then we may use the Euclidean topology and we can study singular points $p$ in arbitrary small $\varepsilon$-neighbourhoods (as we did already at the end of Section A.8). But then we must also allow more functions, since the regular functions at $p$ (in the sense of Definition A.6.1) are always defined in a Zariski neighbourhood of $p$. Thus, instead of considering germs of regular functions at $p$, we consider germs of complex analytic functions at $p=\left(p_1, \ldots, p_n\right) \in \mathbb{C}$. The ring of these functions is isomorphic to the ring of convergent power series $\mathbb{C}\left{x_1-p_1, \ldots, x_n-p_n\right}$, which is a local ring and contains the ring $\mathbb{C}\left[x_1, \ldots, x_n\right]{\left\langle x_1-p_1, \ldots, x_n-p_n\right\rangle}$ of regular functions at $p$.
For arbitrary (algebraically closed) fields $K$, we cannot talk about convergence and then a substitute for $\mathbb{C}\left{x_1-p_1, \ldots, x_n-p_n\right}$ is the formal power series ring $K\left[\left[x_1-p_1, \ldots, x_n-p_n\right]\right]$. Unfortunately, with formal power series, we cannot go into a neighbourhood of $p$; formal power series are just not defined there. Therefore, when talking about geometry of singularities, we consider $K=\mathbb{C}$ and convergent power series. Usually, the algebraic statements which hold for convergent power series do also hold for formal power series (but are easier to prove since we need no convergence considerations). We just mention in passing that there is, for varieties over general fields,another notion of “local” with étale neighbourhoods and Henselian rings (cf. [143]) which is a geometric substitute of convergent power series over $\mathbb{C}$.
For $I \subset \mathbb{C}[x], x=\left(x_1, \ldots, x_n\right)$, an ideal, we have inclusions of rings
$$\mathbb{C}[x] / I \subset \mathbb{C}[x]{\langle x\rangle} / I \mathbb{C}[x]{\langle x\rangle} \subset \mathbb{C}{x} / I \mathbb{C}{x} \subset \mathbb{C}[[x]] / I \mathbb{C}[[x]]$$

## 数学代写|交换代数代写Commutative Algebra代考|Squarefree Factorizatio

Let $K$ be a field of characteristic $p$. In this chapter we will explain how to decompose a univariate polynomial $g \in K[x]$ as a product $g=\prod_{i=1}^k g_{(i)}^i$ of powers of pairwise coprime ${ }^1$ squarefree factors $g_{(1)}, \ldots, g_{(k)}$.

Definition B.1.1. (1) $g \in K[x]$ is called squarefree if $g$ is not constant and if it has no non-constant multiple factor, that is, each irreducible factor of $g$ appears with multiplicity 1 .
(2) Let $g \in K[x], g=\prod_{i=1}^k g_{(i)}^i$ is called the squarefree factorization of $g$ if $g_{(1)}, \ldots, g_{(k)}$ are squarefree, and those $g_{(i)}$, which are non-constant are pairwise coprime.

It follows from the existence and uniqueness of the factorization of $f$ into irreducible factors that the squarefree factorization exists and the squarefree factors are unique up to multiplication by a non-zero constant.

Example B.1.2. Let $g=x^2(x+1)^2(x+3)^4\left(x^2+1\right)^5 \in \mathbb{Q}[x]$ then $g_{(1)}=g_{(3)}=1$ and $g_{(2)}=x(x+1), g_{(4)}=x+3, g_{(5)}=x^2+1$.

As the case of char $K=0$ is an easy exercise using some of the same ideas as in the following proposition we concentrate from this point on on the case of fields of positive characteristic.

Proposition B.1.3. Let $f \in \mathbb{F}_q[x]$ be non-constant with $q=p^r$ and $p$ prime. Then $f$ is squarefree if and only if $f^{\prime} \neq 0$ and $\operatorname{gcd}\left(f, f^{\prime}\right)=1$.

Proof. If $f^{\prime}=0$ then $f=\sum_{j=0}^s a_j x^{p j}$. Since $p$-th roots exist in $\mathbb{F}_q$, i.e. $a_j=b_j^p$ for suitable $b_j \in \mathbb{F}_q$, this implies $f=\sum b_j^p x^{p j}=\left(\sum b_j x^j\right)^p$.

Let $f^{\prime} \neq 0$ and $h$ an irreducible polynomial with $h \mid \operatorname{gcd}\left(f, f^{\prime}\right)$. Then $f=$ $h \cdot g$ and $h \mid\left(f^{\prime}=h^{\prime} g+h g^{\prime}\right)$. If $h^{\prime}=0$ then $h$ is a $p$-th power and hence $f$ is not squarefree. Otherwise $h \mid\left(h^{\prime} g\right)$ and, since $h$ is irreducible, $h \mid g$. This implies $h^2 \mid f$ and again $f$ is not squarefree. Conversely, if $f=h^2 \cdot g$ then $h \mid f^{\prime}$ and $\operatorname{gcd}\left(f, f^{\prime}\right) \neq 1$

## 数学代写|交换代数代写Commutative Algebra代考|Singularities

$$\mathbb{C}[x] / I \subset \mathbb{C}[x]{\langle x\rangle} / I \mathbb{C}[x]{\langle x\rangle} \subset \mathbb{C}{x} / I \mathbb{C}{x} \subset \mathbb{C}[[x]] / I \mathbb{C}[[x]]$$

## 数学代写|交换代数代写Commutative Algebra代考|Squarefree Factorizatio

B.1.1.定义(1)当$g$不为常数且不存在非常数多因子时，称$g \in K[x]$为无平方因子，即$g$的每个不可约因子均以1的倍数出现。
(2)设$g \in K[x], g=\prod_{i=1}^k g_{(i)}^i$为$g$的无平方分解，如果$g_{(1)}, \ldots, g_{(k)}$为无平方分解，非常数的$g_{(i)}$为成对互素数分解。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。