## 数学代考|代数几何代写algebraic geometry代考|Sheaves and Schemes

statistics-lab™ 为您的留学生涯保驾护航 在代写代数几何algebraic geometry方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写代数几何algebraic geometry代写方面经验极为丰富，各种代写代数几何algebraic geometry相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代考|代数几何代写algebraic geometry代考|Sheaves Revisited

Recall that a sheaf of sets $\mathcal{F}$ can be defined on a topological space $X$. It assigns to every open set $U \subseteq X$ the set of sections $\mathcal{F}(U)$. The following properties are required:

1. Restriction: For $V \subseteq U$, we have a restriction map
$$\mathcal{F}(U) \rightarrow \mathcal{F}(V)$$

The restriction is required to be transitive (i.e. for $W \subseteq V \subseteq U$, restriction from $\mathcal{F}(U)$ to $\mathcal{F}(V)$ and then to $\mathcal{F}(W)$ is the same thing as restricting to $\mathcal{F}(W)$ directly). Also, the restriction from $\mathcal{F}(U)$ to itself is just the identity.

1. Gluing: If we have sections $s_{i} \in \mathcal{F}\left(U_{i}\right)$ where $U_{i}$ are open sets, such that $s_{i}$ and $s_{j}$ restrict to the same section in $\mathcal{F}\left(U_{i} \cap U_{j}\right)$, then there exists a unique section $s \in$ $\mathcal{F}\left(\bigcup U_{i}\right)$ which restricts to all the functions $s_{i}$.

The stalk $\mathcal{F}_{X}$ of a sheaf $\mathcal{F}$ at a point $x \in X$ is the set of equivalence classes of sections in $\mathcal{F}(U)$ with $x \in U$ where $U$ is any open set containing $x$, where two sections $s \in \mathcal{F}(U)$, $t \in \mathcal{F}(V)$, are equivalent if they restrict to the same section in $\mathcal{F}(U \cap V)$.

We can also have sheaves of algebraic structures such as groups, abelian groups or rings defined analogously except that $\mathcal{F}(U)$ are groups, abelian groups or rings, and restrictions are homomorphisms.

A morphism of sheaves $\phi: \mathcal{F} \rightarrow \mathcal{G}$ gives for an open set $U$ a map (resp. homomorphism of whatever algebraic structures we are considering)
$$\phi(U): \mathcal{F}(U) \rightarrow \mathcal{G}(U)$$
such that $\phi$ of a restriction of a section $s$ is the restriction of $\phi(s)$.
A morphism of sheaves $\phi: \mathcal{F} \rightarrow \mathcal{G}$ induces, for every $x \in X$, a map (or homomorphism of whatever algebraic structures we have) $\phi_{x}: \mathcal{F}{x} \rightarrow \mathcal{G}{x}$.

If $f: X \rightarrow Y$ is a continuous map and $\mathcal{F}$ is a sheaf on $X$, we have a sheaf $f_{*} \mathcal{F}$ (sometimes called the pushforward) on $Y$ where
$$f_{\circledast} \mathcal{F}(U)=\mathcal{F}\left(f^{-1}(U)\right)$$
for every open set $U \subseteq Y$.

## 数学代考|代数几何代写algebraic geometry代考|Ringed Spaces and Locally Ringed Spaces

Recall that, unless otherwise specified, by a ring, we mean a commutative ring. A ringed space is a topological space $X$ with a sheaf of rings $\mathcal{O}{X}$ (called the structure sheaf). A morphism of ringed spaces $f: X \rightarrow Y$ is a continuous map together with a morphism of sheaves of rings $$\phi: \mathcal{O}{Y} \rightarrow f_{*} \mathcal{O}{X}$$ A locally ringed space is a ringed space where every stalk $\mathcal{O}{X, x}=\left(\mathcal{O}{X}\right){X}$ is a local ring. (Recall that a local ring is a ring which has a unique maximal ideal; a maximal ideal of a ring $R$ is an ideal $m \neq R$ such that there exists no ideal $I$ with $m \subseteq I \subsetneq R$. Equivalently, an ideal $m$ is maximal if and only if $R / m$ is a field.)

A morphism of locally ringed spaces $f: X \rightarrow Y$ is a morphism of ringed spaces such that for every point $x \in X$,
$$\mathcal{O}{Y, f(x)} \stackrel{\phi{f(x)}}{\longrightarrow}\left(f_{*} \mathcal{O}{X}\right){f(x)} \longrightarrow \mathcal{O}{X, x}$$ is a morphism of local rings (where the second map is defined in the obvious way). Here by a morphism of local rings $\phi: R \rightarrow S$ where the maximal ideal of $R$ is $m$ and the maximal ideal of $S$ is $n$, we mean a homomorphism of rings such that $\phi^{-1}(n)=m$ or, equivalently, $\phi(m) \subseteq n$ (see Exercise 2); an example of a homomorphism between local rings which is not a morphism of local rings is the inclusion $\mathbb{Z}{(p)} \subset \mathbb{Q}$, where $\mathbb{Z}_{(p)}$ is $\mathbb{Z}$ localized at the prime ideal ( $p$ ) for $p$ prime, or, in other words, the set of rational numbers whose denominators are not divisible by $p$.)

Note that if $X$ is a locally ringed space and $U \subseteq X$ is an open set, then $U$ with $\mathcal{O}{U}$ equal to the restriction $\left.\mathcal{O}{X}\right|{U}$ of the sheaf $\mathcal{O}{X}$ to $U$ (given by $\left.\mathcal{O}{X}\right|{U}(V)=\mathcal{O}_{X}(V)$ for $V \subseteq U$ open) is a locally ringed space. Let us call it the restriction of the locally ringed space $X$ to $U$.

## 数学代考|代数几何代写algebraic geometry代考|Schemes

An affine scheme is a locally ringed space of the form
$$\operatorname{Spec}(R)={p \mid p \text { is a prime ideal in } R}$$
The topology is the Zariski topology where closed sets are of the form
$$Z_{l}={p \in \operatorname{Spec}(R) \mid I \subseteq p}$$
for an ideal $I$.
As in Chap. 1, we have $Z_{I \cdot J}=Z_{I} \cup Z_{J}$ and
$$Z_{\sum I_{i}}=\bigcap_{i} Z_{I_{i}}$$
thus showing that we have indeed defined a topology. Denote by
$$U_{I}=\operatorname{Spec}(R) \backslash Z_{I}$$
the complementary open set.
A distinguished open set is a set of the form $U_{(r)}$ for $r \in R$ (i.e. $U_{I}$ where $I$ is a principal ideal). Every open set is a union of distinguished open sets. The structure sheaf $\mathcal{O}_{S p e c(R)}$

is uniquely determined by its sections on distinguished open sets by gluing. We have
$$\mathcal{O}{S p e c(R)}\left(U{(r)}\right)=r^{-1} R$$
(recall that $r^{-1} R$ is the set of equivalence classes of fractions $s / r^{n}, s \in R$ by the equivalence relation $s / r^{n} \sim t / r^{m}$ when $r^{n+k} t=r^{m+k} s$ for some $k=0,1,2, \ldots$ ). It is possible to use (1.3.1) as a definition. Some consistency checks are needed. We prefer, however, a definition using actual functions; using our definition, we will prove (1.3.1) in Sect. $2.2$ below (see Lemma 2.2.2).

More concretely, recall that for a commutative ring $R$ and a prime ideal $p$, the localization $R_{p}$ of $R$ at $p$ is the set of equivalence classes
$${r / s \mid r, s \in R, s \notin p} / \sim$$
where
$$\frac{r_{1}}{s_{1}} \sim \frac{r_{2}}{s_{2}}$$
when
$r_{1} s_{2} u=r_{2} s_{1} u$ for some $u \notin p .$

## 数学代考|代数几何代写algebraic geometry代考|Sheaves Revisited

1. 限制：对于在⊆在，我们有一个限制图
F(在)→F(在)

1. 胶合：如果我们有部分s一世∈F(在一世)在哪里在一世是开集，这样s一世和sj限制在同一部分F(在一世∩在j), 那么存在一个唯一的部分s∈ F(⋃在一世)仅限于所有功能s一世.

φ(在):F(在)→G(在)

F⊛F(在)=F(F−1(在))

## 数学代考|代数几何代写algebraic geometry代考|Schemes

（回想起那个r−1R是分数的等价类的集合s/rn,s∈R由等价关系s/rn∼吨/r米什么时候rn+ķ吨=r米+ķs对于一些ķ=0,1,2,…）。可以使用 (1.3.1) 作为定义。需要进行一些一致性检查。然而，我们更喜欢使用实际函数的定义；使用我们的定义，我们将在 Sect. 中证明 (1.3.1)。2.2下面（见引理 2.2.2）。

r/s∣r,s∈R,s∉p/∼

r1s1∼r2s2

r1s2在=r2s1在对于一些在∉p.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代考|代数几何代写algebraic geometry代考|Primary Decomposition

statistics-lab™ 为您的留学生涯保驾护航 在代写代数几何algebraic geometry方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写代数几何algebraic geometry代写方面经验极为丰富，各种代写代数几何algebraic geometry相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代考|代数几何代写algebraic geometry代考|Primary Decomposition

An ideal $q$ in a ring $R$ is called primary if $q \neq R$, and whenever $x y \in q$, we have either $x \in q$ or $y^{n} \in q$ for some $n \in \mathbb{N}$. This definition may seem unnatural at first, because of its asymmetry. In particular, it is not the same thing as a power of a prime ideal (see Exercises 27 and 28). It is, however, obviously true that for a primary ideal $q$, the radical $p=\sqrt{q}$ is prime. We often call $q$ a $p$-primary ideal.

It turns out that the concept of a primary ideal behaves better than many similar notions. Perhaps it could be motivated by noting that an ideal $q$ is primary if and only if in the ring $R / q$, every zero divisor $x$ (which, recall, means a non-zero element $x$ for which there is a nonzero element $y$ with $x y=0$ ) is nilpotent (i.e. satisfies $x^{n}=0$ for some $n \in \mathbb{N}$ ).
Note that this implies the following
5.1.1 Lemma If $q$ is an ideal in a ring $R$ such that $m=\sqrt{q}$ is a maximal ideal, then $q$ is $m$-primary.

Proof Every element of the image $\bar{m}$ of $m$ in $R / q$ is, by assumption, nilpotent. Therefore $\bar{m}=\operatorname{Nil}(R / q)=J a c(R / q)$. Therefore, the ring $R / q$ is local, and every element not in $\bar{m}$ is a unit, and hence cannot be a zero divisor. Thus, every zero divisor in $R / q$ is nilpotent, as we needed to prove.

To further demonstrate the utility of primary ideals, consider the concept of decomposition of ideals: A decomposition of an ideal $I \neq R$ in a ring $R$ is an expression of the form
$$I=J_{1} \cap \cdots \cap J_{n}$$
where $J_{1}, \ldots, J_{n} \neq R$ are ideals. An ideal $I \neq R$ is called indecomposable if it cannot be expressed as $I=J \cap K$ for ideals $J, K \supsetneq I$. Recall that since a Noetherian ring satisfies the ascending chain condition $(\mathrm{ACC}$ ) with respect to ideals, there cannot be an infinite sequence of ideals
$$I_{1} \subsetneq I_{2} \subsetneq I_{3} \ldots \ldots$$

## 数学代考|代数几何代写algebraic geometry代考|Artinian Rings

A commutative ring $R$ is called Artinian if its ideals satisfy the descending chain condition (or DCC), i.e. if every sequence of ideals
$$I_{1} \supseteq I_{2} \supseteq \ldots$$
in $R$ is eventually constant. We will see that this is actually a very restrictive condition (more so than the ACC for ideals).
5.2.1 Lemma An integral domain $R$ which is Artinian is a field.
Proof Let $0 \neq x \in R$. Then by the DCC, $\left(x^{n}\right)=\left(x^{n+1}\right)$ for some $n \in \mathbb{N}$. Therefore, $x^{n}$ is a multiple of $x^{n+1}$, and since $R$ is an integral domain, $x$ is a unit.

Since a quotient of an Artinian ring is obviously Artinian, every prime ideal in an Artinian ring $R$ is maximal, and hence $\operatorname{dim}(R)=0$. Also, obviously, $R$ satisfies (4.3.3).
5.2.2 Lemma The nilradical of an Artinian ring $R$ is nilpotent, i.e. there exists $a k \mathbb{N}$ such that $\operatorname{Nil}(R)^{k}=0$.

Proof By the DCC, there is some $k \in \mathbb{N}$ such that $a=\operatorname{Nil}(R)^{k}=\operatorname{Nil}(R)^{k+1}$. We will show that $a=0$. Assume this is false. Note that then $a \cdot a \neq 0$, i.e. there exists an element $x \in a$ with $x \cdot a \neq 0$. By the DCC, we may further assume that if this is also true with $x$ replaced by $x y$ for some $y \in R$, then $(x)=(x y)$. But now if $x \cdot a \neq 0$, then $x \cdot a \cdot a=x \cdot a \neq 0$, so indeed, there exists a $y \in a$ such that $x y \cdot a \neq 0$. Thus, $(x)=(x y)$, and inductively, $(x)=\left(x y^{n}\right)$ for every $n \in \mathbb{N}$. However, $y$ is by assumption nilpotent, and hence $x=0$, which is a contradiction.
5.2.3 Proposition An Artinian ring $R$ is a product of finitely many local Artinian rings.
(Note that since a product of finitely many Artinian rings is obviously Artinian, this is an if and only if condition.)

Proof By Lemma 5.2.1, every prime ideal of $R$ is maximal. Thus, Nil( $R$ ) is an intersection of maximal ideals. By the DCC, it is an intersection of finitely many maximal ideals:
$$\operatorname{Nil}(R)=m_{1} \cap \cdots \cap m_{n} .$$

## 数学代考|代数几何代写algebraic geometry代考|Dimension

Let $A$ be a Noetherian local ring with maximal ideal $m$ and an $m$-primary ideal $q$ with $s$ generators. We are interested in studying powers of the ideal $q$, but for inductive purposes, a more general concept must be introduced. Let $M$ be a finitely generated $A$-module. A $q$-stable filtration on $M$ is a sequence $\mathcal{M}$ of submodules
$$M=M_{0} \supseteq M_{1} \supseteq M_{2} \supseteq M_{3} \supseteq \ldots$$
such that
$$q M_{i} \subseteq M_{i+1}$$
for all $i \in \mathbb{N}$, and there exists a $k$ such that equality arises for all $i \geq k$. The key point about $q$-stable filtrations is the following
5.3.1 Lemma (Artin-Rees Lemma) Let $M$ be a finitely generated A-module with a $q$ stable filtration $\mathcal{M}$, and let $N \subseteq M$ be a submodule. Then the submodules $N_{i}=M_{i} \cap N$ form a q-stable filtration on $N($ denoted by $\mathcal{M} \cap N)$.
Proof Consider the ring
$$A^{}=\bigoplus_{i \in \mathbb{N}_{0}} q^{i}$$ (where we set $q^{0}=A$ ). The ring structure is by the product from $q^{i}$ and $q^{j}$ to $q^{i+j}$. Then the ring $A^{}$ is a finitely generated $A$-algebra, and hence is Noetherian by the Hilbert basis

theorem, and
$$M^{}=\bigoplus_{i \in \mathbb{N}{0}} M{i}$$
is a finitely generated module (since the filtration on $M$ is $q$-stable). Now consider the submodules
$$N_{k}^{}=\bigoplus_{i \leq k}\left(N \cap M_{k}\right) \oplus \bigoplus_{j \in \mathbb{N}} q^{j}\left(N \cap M_{k}\right) .$$
We have $N_{k}^{} \subseteq N_{k+1}^{}$, so by the $\mathrm{ACC}$, equality arises for large enough $k$, which is what we were trying to prove.

We are interested in measuring the growth of the $A$-modules $M / M_{k}$. Since the ring $A / q$ is Artinian, the finitely generated $A / q$-modules $M_{i} / M_{i+1}$ have finite length, and hence the $A$-modules $M / M_{k}$ have finite length. We put
$$\chi_{q}^{\mathcal{M}}(k)=\ell\left(M / M_{k}\right) .$$

## 数学代考|代数几何代写algebraic geometry代考|Primary Decomposition

5.1.1 引理 Ifq是环中的理想R这样米=q是一个极大理想，那么q是米-基本的。

## 数学代考|代数几何代写algebraic geometry代考|Artinian Rings

5.2.1 引理 积分域R这是 Artinian 是一个领域。

5.2.2 引理 Artinian 环的零根R是幂零的，即存在一种ķñ这样零⁡(R)ķ=0.

5.2.3 命题 Artinian 环R是有限多个局部 Artinian 环的乘积。
（请注意，由于有限多个 Artinian 环的乘积显然是 Artinian，因此这是一个当且仅当条件。）

## 数学代考|代数几何代写algebraic geometry代考|Dimension

q米一世⊆米一世+1

5.3.1 引理 (Artin-Rees Lemma) Let米是一个有限生成的 A 模q稳定过滤米， 然后让ñ⊆米成为一个子模块。然后是子模块ñ一世=米一世∩ñ形成一个q-稳定的过滤ñ(表示为米∩ñ).

ñķ=⨁一世≤ķ(ñ∩米ķ)⊕⨁j∈ñqj(ñ∩米ķ).

χq米(ķ)=ℓ(米/米ķ).

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代考|代数几何代写algebraic geometry代考| Computing with Polynomials

statistics-lab™ 为您的留学生涯保驾护航 在代写代数几何algebraic geometry方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写代数几何algebraic geometry代写方面经验极为丰富，各种代写代数几何algebraic geometry相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代考|代数几何代写algebraic geometry代考|Divisibility of Polynomials

We begin with some very basic facts about divisibility.
4.1.1 Proposition (Chinese Remainder Theorem) Let $I_{1}, \ldots, I_{n}$ be ideals in a commutative ring $R$ such that $1 \in I_{i}+I_{j}$ for all $i \neq j$. Then $I_{1} \cap \cdots \cap I_{n}=I_{1} \cdots \cdots I_{n}$ (the product of ideals is the ideal generated by $x_{1} \cdots x_{n}$ with $\left.x_{i} \in I_{i}\right)$ and the product of

projections
$$R /\left(I_{1} \cap \cdots \cap I_{n}\right) \rightarrow \prod_{i=1}^{n} R / I_{i}$$
is an isomorphism.
Proof It suffices to consider the case $n=2$ (then we can use induction). For $n=2$, we always have $I_{1} I_{2} \subseteq I_{1} \cap I_{2}$. To show the opposite inequality, let $1=a_{1}+a_{2}$ where $a_{i} \in I_{i}$. Then for $x \in I_{1} \cap I_{2}, x=x a_{1}+x a_{2} \in I_{1} I_{2}$. Now (4.1.1) is always injective since an element goes to 0 on the right hand side if and only if it is in every $I_{i}$. To show surjectivity for $n=2$, choosing $x_{1}, x_{2} \in R$, the element $x_{1} a_{2}+x_{2} a_{1}$ is congruent to $x_{i}$ modulo $I_{i}$ for $i=1,2$, which proves surjectivity.

An element $u \in R$ of a commutative ring is called a unit if there exists another element $u^{-1} \in R$ such that $u u^{-1}=1$. Let $R$ be an integral domain. An irreducible element is an element $x \in R$ which is not zero or a unit such that $y z=x$ implies that one of the elements $y, z$ is a unit. An integral domain $R$ is called a unique factorization domain (or UFD) if every element $x \in R$ which is not 0 or a unit factors uniquely into irreducible elements up to order and multiplication by units, i.e.
$$x=x_{1} \ldots x_{n}$$
where $x_{i}$ are irreducible, and whenever
$$x=y_{1} \ldots y_{m}$$
where $y_{i}$ are irreducible, we have $m=n$ and there exists a permutation $\sigma$ and units $u_{i}$ such that
$$x_{i}=u_{i} y_{\sigma(i)}$$
In a UFD, any set of elements $S$ has a greatest common divisor (GCD) which is an element $x$ dividing all elements of $S$ such that every other elements dividing all elements of $S$ divides $x$. The GCD is, of course, uniquely determined up to multiplication by a unit.
A particular type of example of a UFD is a principal ideal domain (or PID) which means an integral domain whose every ideal is principal (i.e. generated by a single element). In particular, then $R$ is Noetherian, which guarantees that a decomposition into irreducible elements exists. Then the principal ideal property guarantees that an irreducible element $a$ generates a prime ideal: if $x y \in(a)$ and $x \notin(a)$, then $(x, a)=(b)$ for some element $b$, but $b$ has to be a unit by irreducibility. Thus, $y \in(y x, y a) \subseteq(a)$.

## 数学代考|代数几何代写algebraic geometry代考|Gröbner Basis

We shall now prove that rings of polynomials over a Noetherian ring are Noetherian. In the special case of multivariable polynomials over a field, we can be a lot more explicit, with computational applications.
4.2.1 Theorem (Hilbert Basis Theorem) If a ring $R$ is a Noetherian, then so is the ring of polynomials $R[x]$.

COMMENT In this context, the term “basis” refers to a set of generators of an ideal, no linear independence is implied.

Proof Assume $R$ is Noetherian. Let $I \subseteq R[x]$ be an ideal. Then the top coefficients (i.e. coefficients of the highest power of $x$ ) of all the polynomials $f \in I$ form an ideal $J \subseteq R$ (since two nonzero polynomials of unequal degrees can be brought to the same degree by multiplying the polynomial of lesser degree by a power of $x$ ). By assumption, then, the ideal $J$ is generated by the top coefficients of some polynomials $f_{1}, \ldots, f_{n} \in I$.

Let $d$ be the maximum of the degrees of the polynomials $f_{1}, \ldots, f_{n}$. Then by construction, for any polynomial $g \in I$ of degree $\geq d$, there exist $a_{1}, \ldots a_{n} \in R$, $m_{1} \ldots, m_{n} \in \mathbb{N}{0}$ such the top coefficients of $g(x)$ and $a{1} f_{1}(x) x^{m_{1}}+\ldots a_{n} f_{n}(x) x^{m_{n}}$ coincide. By induction, then, there exists an $R[x]$-linear combination $g_{0}$ of the polynomials $f_{1}, \ldots, f_{n}$ such that $g(x)-g_{0}(x)$ is either 0 or is of degree $<d$.

Now consider for each fixed $i \in \mathbb{N}{0}$ the ideal $J{i} \subseteq R$ of all the top coefficients of all polynomials in $I$ of degree $i$. Then each of these ideals $J_{i}$ is finitely generated, so by taking finitely many polynomials $h_{1}, \ldots, h_{\ell}$ in $I$ of degrees $i=0, \ldots, d-1$ whose top coefficients are the generators of all the $J_{i}$ ‘s, $0 \leq i<d$, we see that every polynomial in $I$ of degree $<d$ is an $R$-linear combination of $h_{1}, \ldots, h_{\ell}$. Thus, we are done.

Next, we shall discuss the ring $k\left[x_{1}, \ldots, x_{n}\right]$ of polynomials in $n$ variables over a field $k$. Even though this ring is not a Euclidean domain for $n>1$ (because it is not a PIDthink, for example, of the ideal $\left(x_{1}, \ldots, x_{n}\right)$ ) there is a certain analog of the long division algorithm which allows us decide, for example, whether a polynomial is an element of a given ideal, or whether two ideals are the same.

By a monomial, we shall mean an expression of the form $x_{1}^{m_{1}} \ldots x_{n}^{m_{n}}$, i.e. equivalently, the $n$-tuple $a=\left(m_{1}, \ldots, m_{n}\right) \in \mathbb{N}{0}^{n}$, which are sometimes referred to as multidegrees. For what follows, we need to fix a monomial order. This means a total ordering $\geq$ on $n$-tuples of non-negative integers (i.e. for any two $n$-tuples $a, b$ we have $a \geq b$ or $b \geq a$ ) which satisfies the descending chain condition (or DCC), i.e. any sequence $a{1} \geq a_{2} \geq \ldots$ is eventually constant. A totally ordered set satisfying the DCC is also sometimes called well ordered. In addition, we require that for multidegrees $a, b$, $c$, if $a \geq b$, then $a+c \geq b+c$.
Note that this implies that the multidegree $(0, \ldots, 0)$ is the smallest (since otherwise, the DCC would be violated). This implies that when $m_{i} \leq p_{i}$ for all $i=1, \ldots n$, then $\left(m_{1}, \ldots, m_{n}\right) \leq\left(p_{1}, \ldots, p_{n}\right)$

## 数学代考|代数几何代写algebraic geometry代考|Nullstellensatz

4.3.2 Proposition Suppose $k$ is an algebraically closed field. Then every maximal ideal $I \subset k\left[x_{1}, \ldots, x_{n}\right]$ is of the form
$$\left(x_{1}-a_{1}, \ldots, x_{n}-a_{n}\right)$$
for some $a_{1}, \ldots, a_{n} \in k$.
Proof We will show that any ideal $I \subseteq k\left[x_{1}, \ldots, x_{n}\right]$ is contained in an ideal of the form (4.3.1). This is proved by induction on $n$. Suppose the statement is true with $n \geq 1$ replaced by any lower number. (For $n=1$, the assumption is vacuous.) Then there are two possibilities:

Case 1: The ideal $J=I \cap k\left[x_{n}\right]$ in $k\left[x_{n}\right]$ is non-zero. Then, since $k\left[x_{n}\right]$ is a PID, $J=$ $(f)$ is a principal ideal, and since $k$ is algebraically closed, $f$ factors into powers of linear factors $\left(x_{n}-b_{i}\right)^{\ell_{i}}, i=1, \ldots, m$. By the Chinese Remainder Theorem, $k\left[x_{1}, \ldots, x_{n}\right] / I$ is isomorphic to the product of the rings $k\left[x_{1}, \ldots, x_{n}\right] /\left(I+\left(x_{n}-b_{i}\right)^{\ell_{i}}\right)$. Then, for some $i$, $I+\left(x_{n}-b_{i}\right)^{\ell_{i}} \neq(1)$, but this implies $I+\left(x_{n}-b_{n}\right) \neq(1)$ (since an ideal whose radical is (1) is itself (1)). Therefore, we can pass to the ring $k\left[x_{1}, \ldots, x_{n}\right] /\left(x_{n}-b_{i}\right) \cong k\left[x_{1}, \ldots, x_{n-1}\right]$ and use the induction hypothesis.

Case $2: I \cap k\left[x_{n}\right]=(0)$. Therefore, if we set $R=k\left(x_{n}\right)\left[x_{1}, \ldots, x_{n-1}\right]$ (recall that $k(x)$ denotes the field of rational functions in $k$ in one variable), then $I, R \neq R$. Now we can apply the induction hypothesis to the ring of polynomials $\bar{R}=\overline{k\left(x_{n}\right)}\left[x_{1}, \ldots, x_{n-1}\right]$ where $\overline{k\left(x_{n}\right)}$ denotes the algebraic closure of $k\left(x_{n}\right)$. Thus, the ideal $I \cdot \bar{R}$ is contained in an ideal of the form $\left(x_{1}-b_{1}, \ldots, x_{n-1}-b_{n-1}\right)$ for $b_{i} \in \overline{k\left(x_{n}\right)}$. Thus, each $b_{i}$ is the root of a polynomial with coefficients in $k\left(x_{n}\right)$. Now since $k$ is algebraically closed, all of the coefficient polynomials factor into linear factors, and there are only finitely many values of $x_{n} \in k$ for which either the denominator or numerator of any of the coefficient polynomials is 0 . Since $k$ is algebraically closed, it is infinite, and we can choose an element $a_{n} \in k$ which is different from any of those values. Plugging in $x_{n}=a_{n}$, all the expressions for $b_{i}$ give meaningful formulas for elements $a_{i} \in k$. Then, the ideal $I$ is contained in (4.3.1).

## 数学代考|代数几何代写algebraic geometry代考|Divisibility of Polynomials

4.1.1 命题（中国剩余定理）让一世1,…,一世n是交换环中的理想R这样1∈一世一世+一世j对全部一世≠j. 然后一世1∩⋯∩一世n=一世1⋯⋯一世n（理想的乘积是由X1⋯Xn和X一世∈一世一世)和产品

R/(一世1∩⋯∩一世n)→∏一世=1nR/一世一世

X=X1…Xn

X=是1…是米

X一世=在一世是σ(一世)

UFD 的一个特定类型的示例是主理想域（或 PID），这意味着一个积分域，其每个理想都是主域（即由单个元素生成）。特别是，那么R是 Noetherian，它保证存在分解成不可约元素。那么主理想性质保证一个不可约元素一种产生一个素理想：如果X是∈(一种)和X∉(一种)， 然后(X,一种)=(b)对于某些元素b， 但b必须是不可约的单位。因此，是∈(是X,是一种)⊆(一种).

## 数学代考|代数几何代写algebraic geometry代考|Gröbner Basis

4.2.1 定理（希尔伯特基定理）如果环R是 Noetherian，那么多项式环也是R[X].

## 数学代考|代数几何代写algebraic geometry代考|Nullstellensatz

4.3.2 命题假设ķ是代数闭域。那么每个极大理想一世⊂ķ[X1,…,Xn]是形式
(X1−一种1,…,Xn−一种n)

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代考|代数几何代写algebraic geometry代考| Rational Maps, Smooth Maps and Dimension

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代考|代数几何代写algebraic geometry代考|Definition of a Rational Map

For two algebraic varieties $X, Y$, a rational map from $X$ to $Y$ is morphism of varieties
$$f: U \rightarrow Y$$
where $U$ is a non-empty Zariski open subset of $X$. The rational map $f$ is considered equal to a rational map
$$g: V \rightarrow Y$$
if
$$f(x)=g(x) \text { for all } x \in U \cap V$$
Recall that (by irreducibility), a non-empty Zariski open set in a variety $X$ is dense which means that its complement does not contain any non-empty open set. This implies that for non-empty Zariski open sets $U, V \subseteq X, U \cap V$ is non-empty and Zariski open.

A rational map $f: X \rightarrow Y$ is called dominant if for $W \subseteq Y$ non-empty Zariski open, the inverse image $f^{-1}(W)$ is non-empty (note that it is by definition open).

## 数学代考|代数几何代写algebraic geometry代考|The Category of Varieties and Dominant Rational Maps

One can consider the category whose objects are varieties, and morphisms are dominant rational maps. An isomorphism in this category is called a birational equivalence. A variety is called rational if it is birationally equivalent to an affine (equivalently, projective) space. Note also that any rational map of varieties that has an inverse as a rational map is necessarily dominant. This means that if we consider the larger category of varieties and all rational maps (not necessarily dominant), it has the same isomorphisms as the category of varieties and rational dominant maps.
$$X \mapsto K(X)$$
where $X$ is a variety and $K(X)$ is its field of rational functions gives rise to an equivalence of categories between the category of varieties and rational dominant maps, and the opposite of the category of fields containing $\mathbb{C}$ which are finitely generated (as fields) over $\mathbb{C}$, and homomorphisms of $\mathbb{C}$-algebras (or, equivalently, homomorphisms of fields which fix $\mathbb{C}$ ). Such fields are sometimes known as function fields over $\mathbb{C}$. Functoriality follows from functoriality of the field of fractions with respect to injective homomorphisms of integral domains.

To go the other way, select, for a function field $K$ over $\mathbb{C}$, elements $x_{1}, \ldots x_{n}$ which generate $K$ as a field containing $\mathbb{C}$. This defines a map
$$h: \mathbb{C}\left[x_{1}, \ldots, x_{n}\right] \rightarrow K$$
Then send $K$ to $Z(I)$ where $I$ is the kernel of $h$, i.e. the ideal of all polynomials $p$ such that $p(h)=0$. This is an affine variety since the ideal $I$ is prime (because the quotient by $I$, which is the image of $h$, is an integral domain). By definition, homomorphisms of fields give rise to rational maps, and the two constructions are inverse to each other.
In particular, a variety $X$ is rational if and only if
$$K(X) \cong \mathbb{C}\left(x_{1}, \ldots, x_{n}\right)$$
as $\mathbb{C}$-agebras for some $n$. (Note: the right hand side means the field of rational functions on $A^{n}$, i.e. the field of fractions of the ring of polynomials $\mathbb{C}\left[x_{1}, \ldots, x_{n}\right]$.) We also say that the field $K(X)$ is rational.

## 数学代考|代数几何代写algebraic geometry代考|Standard Smooth Homomorphisms of Commutative Rings

A homomorphism of commutative rings
$$f: A \rightarrow B$$
is called standard smooth of dimension $k \geq 0$ if $f$ can be expressed as
$$A \rightarrow A\left[x_{1}, \ldots, x_{n}\right] /\left(f_{1}, \ldots, f_{m}\right) \cong B$$
where $n=m+k$, the first homomorphism sends $a \in A$ to $a$, and $f_{i}$ are polynomials such that the ideal in
$$A\left[x_{1}, \ldots, x_{n}\right] /\left(f_{1}, \ldots, f_{m}\right)$$
generated by the determinants of the $m \times m$ submatrices of the Jacobi matrix
$$\left(\begin{array}{ccc} \frac{\partial f_{1}}{\partial x_{1}} & \cdots & \frac{\partial f_{1}}{\partial x_{n}} \ \cdots & \cdots & \cdots \ \frac{\partial f_{m 1}}{\partial x_{1}} & \cdots & \frac{\partial f_{m}}{\partial x_{n}} \end{array}\right)$$
is $A\left[x_{1}, \ldots, x_{n}\right] /\left(f_{1}, \ldots, f_{m}\right)$ (or, equivalently, contains 1$)$. This is equivalent to saying that the ideal in $A\left[x_{1}, \ldots, x_{n}\right]$ generated by the determinants and $f_{1}, \ldots, f_{m}$ contains 1 . If $A$ is a field, this can be tested using Gröbner basis algorithm, which we will learn in the next Section.

## 数学代考|代数几何代写algebraic geometry代考|Definition of a Rational Map

F:在→是

G:在→是

F(X)=G(X) 对全部 X∈在∩在

X↦ķ(X)

H:C[X1,…,Xn]→ķ

ķ(X)≅C(X1,…,Xn)

## 数学代考|代数几何代写algebraic geometry代考|Standard Smooth Homomorphisms of Commutative Rings

F:一种→乙

(∂F1∂X1⋯∂F1∂Xn ⋯⋯⋯ ∂F米1∂X1⋯∂F米∂Xn)

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代考|代数几何代写algebraic geometry代考|The Category of Topological Spaces

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代考|代数几何代写algebraic geometry代考|The Category of Algebraic Varieties

Recall that at this moment, our definition of an algebraic variety includes affine, quasiaffine, projective and quasiprojective varieties, defined in Sects. 1.3.1 and 1.3.2. These are the objects of the category of algebraic varieties. Following our general guiding principle, morphisms of algebraic varieties $f: X \rightarrow Y$ are the mappings which preserve the structure.

The structure consists of the Zariski topology, and regular functions. It is correct to say that morphisms of varieties are those mappings which preserve topology and regular functions.
In more detail, then, morphisms of algebraic varieties
$$f: X \rightarrow Y$$
are maps which

1. are continuous with respect to the Zariski topology
2. have the property that if $g: U \rightarrow \mathbb{C}$ is a regular function, then the composition
$$g \circ f: f^{-1}(U) \rightarrow \mathbb{C}$$

or, more precisely,
$$\left.g \circ f\right|_{f^{-1}(U)}: f^{-1}(U) \rightarrow \mathbb{C},$$
is a regular function. Note the inverse image in both formulas. The extra notation in the second formula means the restriction of a function.

Note that in particular, by the second property, a morphism of varieties $f: X \rightarrow Y$ specifies (we sometimes say: induces) a homomorphism of rings
$$\mathbb{C}[f]: \mathbb{C}[Y] \rightarrow \mathbb{C}[X]$$
In fact, both rings contain $\mathbb{C}$ (thought of as constant functions), and the homomorphism $\mathbb{C}[f]$ fixes $\mathbb{C}$, i.e. satisfies
$$\mathbb{C}f=\lambda \text { for } \lambda \in \mathbb{C}$$
Commutative rings $R$ with a homomorphism of rings
$$A \rightarrow R$$
for some other commutative ring $A$ are called (commutative) A-algebras. Therefore, $C[f]$ is a homomorphism of $\mathbb{C}$-algebras. In general, a homomorphism of $A$-algebras $R \rightarrow R^{\prime}$ is defined as a homomorphism of rings which commutes with the homomorphisms from $A$ to $R, R^{\prime}$.

## 数学代考|代数几何代写algebraic geometry代考|The Morphisms into an Affine Variety

An easy but powerful theorem states that a morphism in the category of varieties
$$f: X \rightarrow Y$$
where $Y$ is affine is characterized by the induced homomorphism of $\mathbb{C}$-algebras
$$\mathbb{C}[X] \leftarrow \mathbb{C}[Y]$$
(note the reversal of the arrow, called contravariance).

This means, in more detail, that for every homomorphism of rings (2.3.2), there exists a unique morphism of varieties $(2.3 .1)$ which induces it, provided that $Y$ is affine. In particular:

The category of affine varieties over $\mathbb{C}$ and morphisms of varieties is equivalent to the opposite category of the category of finitely generated $\mathbb{C}$-algebras and homomorphisms of C-algebras.

To see why $(2.3 .1)$ and $(2.3 .2)$ are equivalent for $Y$ affine, note that the passage from (2.3.1) to $(2.3 .2)$ is immediate from the definition. On the other hand, given a homomorphism
$$\mathbb{C}\left[x_{1}, \ldots, x_{n}\right] / I(Y) \rightarrow \mathbb{C}[X]$$
we can take the images of the generators $x_{1}, \ldots, x_{n}$ as coordinates of a mapping from $X$ to $\mathrm{A}_{\mathrm{C}}^{n}$, which lands in $Y$. It is readily checked that this is a morphism of varieties, and that both passages between (2.3.1) and (2.3.2) are inverse to each other (although note that in our current setting, there are several cases for $X$ to consider!).

Roughly speaking, we think of affine varieties as those which have “enough regular functions.” From this point of view, they are the opposite of projective varieties: The only algebraic variety which is both affine and projective is a single point (and in some definitions, the empty set, but our definition of irreducibility excludes the empty set, so we do not count it).

## 数学代考|代数几何代写algebraic geometry代考|Quasiaffine Varieties which are not Isomorphic to Affine Varieties

The theorem described at the beginning of Sect. $2.3$ can be useful in deciding which varieties are isomorphic to affine varieties. For example, note that in $\mathrm{A}{\mathbb{C}}^{n}$, $${(0, \ldots, 0)}=Z\left(x{1}, \ldots, x_{n}\right)$$
( $Z$ denotes the set of zeros, see Sect. 1.1.2). By (1.4.6), (1.4.2), for $n \geq 2$, we have
\begin{aligned} &\mathbb{C}\left[\mathbb{A}{\mathbb{C}}^{n} \backslash{(0, \ldots, 0)}\right]= \ &=\mathbb{C}\left[x{1}, \ldots, x_{n}\right]\left[x_{1}^{-1}\right] \cap \cdots \cap \mathbb{C}\left[x_{1}, \ldots, x_{n}\right]\left[x_{n}^{-1}\right]= \ &=\mathbb{C}\left[x_{1}, \ldots, x_{n}\right]=\mathbb{C}\left[\mathbb{A}{\mathbb{C}}^{n}\right] \end{aligned} Since we know that $\mathrm{A}{\mathbb{C}}^{n}$ is affine, this means that
$$\mathrm{A}{\mathbb{C}}^{n} \backslash{(0, \ldots 0)}$$ is not affine for $n \geq 2$ : its inclusion into $\mathrm{A}{\mathrm{C}}^{n}$ induces an isomorphism of rings of regular function, but is not an isomorphism of varieties (it is not onto on points), so if both varieties were isomorphic to affine varieties, it would contradict the theorem at the beginning of Sect. 2.3).

## 数学代考|代数几何代写algebraic geometry代考|The Category of Algebraic Varieties

F:X→是

1. 关于 Zariski 拓扑是连续的
2. 拥有如果G:在→C是正则函数，则组成
G∘F:F−1(在)→C

G∘F|F−1(在):F−1(在)→C,

C[F]:C[是]→C[X]

$$\mathbb{C} f =\lambda \text { for } \lambda \in \mathbb{C} C这米米在吨一种吨一世在和r一世nGsR在一世吨H一种H这米这米这rpH一世s米这Fr一世nGs 其他交换环的 \rightarrow R$$

## 数学代考|代数几何代写algebraic geometry代考|The Morphisms into an Affine Variety

F:X→是

C[X]←C[是]
（注意箭头的反转，称为逆变）。

C[X1,…,Xn]/一世(是)→C[X]

## 数学代考|代数几何代写algebraic geometry代考|Quasiaffine Varieties which are not Isomorphic to Affine Varieties

Sect 开头描述的定理。2.3可用于确定哪些变体与仿射变体同构。例如，请注意在一种Cn,(0,…,0)=从(X1,…,Xn)
( 从表示零的集合，参见 Sect. 1.1.2)。由（1.4.6），（1.4.2），对于n≥2， 我们有
C[一种Cn∖(0,…,0)]= =C[X1,…,Xn][X1−1]∩⋯∩C[X1,…,Xn][Xn−1]= =C[X1,…,Xn]=C[一种Cn]既然我们知道一种Cn是仿射的，这意味着

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代考|代数几何代写algebraic geometry代考|Categories, and the Category of Algebraic Varieties

statistics-lab™ 为您的留学生涯保驾护航 在代写代数几何algebraic geometry方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写代数几何algebraic geometry代写方面经验极为丰富，各种代写代数几何algebraic geometry相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代考|代数几何代写algebraic geometry代考|The Definition of a Category, and an Example: The Category of Sets

In a category $C$, we have a class of objects $\operatorname{Obj}(C)$ and a class of morphisms $\operatorname{Mor}(C)$, satisfying certain axioms.

Explaining the need to distinguish between sets and classes takes us on a brief detour into set theory. It comes from the fact that the naive interpretation of the notation
$${X \mid \ldots}$$
as “the set of all sets $X$ such that …” leads to a contradiction in
$${X \mid X \notin X}$$
where neither $X \in X$ nor $X \notin X$ are possible; because of that, we distinguish between sets and classes and interpret (2.1.1) as “the class of all sets $X$ such that …,” and define a set as a class which is an element of another class. Otherwise, it is called a proper class. Note that then (2.1.2) is just an example of a proper class; in fact, it is the class of all sets.
The axioms of a category say that $\operatorname{Obj}(C)$ and $\operatorname{Mor}(C)$ satisfy all the formal properties of the most basic example: the category Sets whose objects are sets and morphisms are mappings of sets. Thus, we have two mappings
$$S, T: \operatorname{Mor}(C) \rightarrow \operatorname{Obj}(C)$$
(called source and target, which in the category of sets are the domain and codomain of a mapping). A morphism $f \in \operatorname{Mor}(C)$ with $S(f)=X, T(f)=Y$ (where $X, Y$ are objects) is called a morphism from $X$ to $Y$, and denoted by
$$f: X \rightarrow Y$$
or
$$X \stackrel{f}{\longrightarrow} Y$$
same as for mappings of sets. We have a mapping $\operatorname{Obj}(C) \rightarrow \operatorname{Mor}(C)$ called the identity morphism
$$I d_{X}: X \rightarrow X$$
Also like for mappings, the structure of a category specifies, for two morphisms
$$f: X \rightarrow Y, g: Y \rightarrow Z,$$
the composition
$$g \circ f: X \rightarrow Z$$

(note the reversal of order of $f$ and $g$, motivated by mappings: when we apply mappings to an element, we write $g \circ f(x)=g(f(x))$, even though we apply $f$ first).

Morphisms, however, may not always be mappings, (although in the category Sets, and many other examples, they are), and so they cannot be, in the context of pure category theory, applied to elements. So instead, we must define a category by axioms. These axioms are simple: they say that the source and target of $I d_{X}$ are equal to $X$, and that the composition of morphisms is associative
$$(h \circ g) \circ f=h \circ(g \circ f)$$
(when applicable) and unital, i.e. for $f: X \rightarrow Y$,
$$I d_{Y} \circ f=f \circ I d_{X}=f$$
Lastly, we require that the class $C(X, Y)$ of all morphisms $f: X \rightarrow Y$ be a set. We call the category $C$ small if the class $O b j(C)$ is a set. (Then necessarily also $M o r(C)$ is a set.)
To see that morphisms do not always have to be mappings, note that to every category $C$, there is the opposite (sometimes also called dual) category $C^{O p}$ which “turns around the arrows”: $O b j\left(C^{O p}\right)=\operatorname{Obj}(C), \operatorname{Mor}\left(C^{O p}\right)=\operatorname{Mor}(C)$ and $I d$ is $C$ and $C^{O p}$ are the same, but $S$ in $C^{O p}$ is $T$ in $C$ and vice versa, and composition of morphisms $\alpha \circ \beta$ in $C^{O p}$ is $\beta \circ \alpha$ in $C$.

## 数学代考|代数几何代写algebraic geometry代考|Categories of Algebraic Structures

One purpose of categories is to be able to discuss, and relate, mathematical structures of the same kind. For example, all sets, all groups, all abelian groups, all rings, all topological spaces, all algebraic varieties. (Recall that a group has one operation which is associative, unital and has an inverse; an abelian group is a group which is also commutative.) So we want a category whose objects are the given structures, i.e. the category of groups, rings, etc. But what should the morphisms be?

Of course, we may be able to define the morphisms in a fairly arbitrary way, as long as they satisfy the axioms of a category, which we learned in Sect. 2.1.1. For example, we could define the only morphisms to be identities, but that would not be very useful for understanding the given mathematical structure. This is why, usually, there is a standard choice of morphisms of mathematical structures of a given kind, which are, vaguely speaking, mappings which preserve the given structure. Making this precise requires different techniques in different cases.

The case which is the easiest to handle are categories of algebraic structures. An algebraic structure comes with operations (example: addition or multiplication). In this case, the default choice of morphisms are homomorphisms of the given algebraic structures, which means mappings which preserve the operations.

For example, a homomorphism of groups $f: G \rightarrow H$, written multiplicatively, is required to satisfy
$$f(x \cdot y)=f(x) \cdot f(y)$$
(Philosophically, the unit and inverse are also operations, so we should include $f(1)=1$ and $f\left(x^{-1}\right)=(f(x))^{-1}$, but in the case of groups, it follows from the axioms.) The category of groups and homomorphisms is denoted by Grp, the category of abelian groups and homomorphisms is denoted by $A b$.
Analogously, a homomorphism of rings satisfies
$$f(x+y)=f(x)+f(y)$$
and
$$f(x y)=f(x) f(y)$$
A non-zero ring is not a group with respect to multiplication (because one cannot divide by 0 ), so we must also require
$$f(1)=1,$$
since it does not follow automatically.
One must be careful not to confuse a homomorphism of rings with a homomorphism of $R$-modules over a fixed ring $R$. (Recall that a module over a commutative ring $R$ is an abelian group $M$ with an operation of taking multiples by elements $r \in R$ which satisfies distributivity from both sides, unitality and associativity; an example of an $R$-module is $R$ itself or more generally an ideal of $R$, which is the same thing as a submodule of the $R$-module $R$.)
Thus, a homomorphism $f: M \rightarrow N$ of $R$-modules satisfies
\begin{aligned} &f(x+y)=f(x)+f(y) \ &f(r x)=r f(x) \text { for } r \in R \end{aligned}
Sometimes, the same algebraic object may be used for two different purposes. For example, as already remarked, a ring $R$ is a module over itself. In such cases, we must be careful to specify which category we are working in.

## 数学代考|代数几何代写algebraic geometry代考|Functors and Natural Transformations

Let $C, D$ be categories. A functor $F: C \rightarrow D$ consists of maps $F=O b j(F)$ : $\operatorname{Obj}(C) \rightarrow \operatorname{Obj}(D), F=\operatorname{Mor}(F): \operatorname{Mor}(C) \rightarrow \operatorname{Mor}(D)$ which preserves identity, source, target and composition: For $X \in O b j(C), f, g \in \operatorname{Mor}(C)$,
$$\begin{gathered} F\left(I d_{X}\right)=I d_{F(X)}, \ F(S(f))=S(F(f)), \ F(T(f))=T(F(f)), \ F(g \circ f)=F(g) \circ F(f) . \end{gathered}$$
when applicable.
A natural transformation $\eta: F \rightarrow G$ is a collection of morphisms
$$\eta_{X}: F(X) \rightarrow G(X), X \in O b j(C),$$
such that for every morphism $f: X \rightarrow Y$ in $C$, we have a commutative diagram:
Commutativity means that the two compositions of arrows (i.e. morphisms) indicated in the diagram are equal.

An equivalence of categories $C, D$ is a pair of functors $F: C \rightarrow D$ and $G: D \rightarrow C$ and natural isomorphisms (i.e. natural transformations which have inverses)
\begin{aligned} &F \circ G \cong I d_{D}, \ &G \circ F \cong I d_{C} . \end{aligned}

## 数学代考|代数几何代写algebraic geometry代考|The Definition of a Category, and an Example: The Category of Sets

X∣…

X∣X∉X

（称为源和目标，在集合的类别中是映射的域和共域）。态射F∈铁道部⁡(C)和小号(F)=X,吨(F)=是（在哪里X,是是对象）称为态射X到是，并表示为
F:X→是

X⟶F是

F:X→是,G:是→从,

G∘F:X→从

（注意顺序颠倒F和G，由映射驱动：当我们将映射应用到一个元素时，我们写G∘F(X)=G(F(X)), 即使我们申请F第一的）。

(H∘G)∘F=H∘(G∘F)
（如适用）和单位，即F:X→是,

## 数学代考|代数几何代写algebraic geometry代考|Categories of Algebraic Structures

F(X⋅是)=F(X)⋅F(是)
（从哲学上讲，单位和逆也是运算，所以我们应该包括F(1)=1和F(X−1)=(F(X))−1，但在群的情况下，它来自公理。）群和同态的范畴由 Grp 表示，阿贝尔群和同态的范畴由一种b.

F(X+是)=F(X)+F(是)

F(X是)=F(X)F(是)

F(1)=1,

F(X+是)=F(X)+F(是) F(rX)=rF(X) 为了 r∈R

## 数学代考|代数几何代写algebraic geometry代考|Functors and Natural Transformations

F(一世dX)=一世dF(X), F(小号(F))=小号(F(F)), F(吨(F))=吨(F(F)), F(G∘F)=F(G)∘F(F).

F∘G≅一世dD, G∘F≅一世dC.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代考|代数几何代写algebraic geometry代考|Regular Functions on an Affine Variety

statistics-lab™ 为您的留学生涯保驾护航 在代写代数几何algebraic geometry方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写代数几何algebraic geometry代写方面经验极为丰富，各种代写代数几何algebraic geometry相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代考|代数几何代写algebraic geometry代考|Regular Functions on an Affine Variety

Regular functions on an affine variety (or, more generally, an affine algebraic set) $X \subseteq \mathbb{A}{C}^{n}$ are also polynomials in the sense that they do not have a denominator. More precisely, we have $$\mathbb{C}[X]=\mathbb{C}\left[x{1}, \ldots, x_{n}\right] / I(X)$$
where $I(X)$ is the ideal of all polynomials which are 0 at every point of $X$. Recall that the division symbol in (1.4.3) denotes cosets, i.e. the elements of the ring (1.4.3) are cosets, which are sets of the form
$$p+I(X)={p+q \mid q \in I(X)}$$
Recall that cosets are the algebraic device for setting the elements of $I(X)$ equal to 0 in the ring (1.4.3), which is what we want, since they are constantly zero as functions on $X$.
To see that (1.4.3) is the correct formula for the ring of regular functions on $X$, first note that the right hand side of (1.4.3) maps injectively into the left hand side by the definition

of the ideal $I(X)$. To show that the map is onto, we need to show that if we cover $X$ with Zariski open sets $U_{i}$ in $\mathbb{A}{C}^{n}$ and exhibit rational functions $g{i} / h_{i}$ where $h_{i} \neq 0$ on $U_{i}, i \in S$, which such that $g_{i} / h_{i}=g_{j} / h_{j}$ on $U_{i} \cap U_{j} \cap X$, then there exists a polynomial $\phi$ which restricts to each $g_{i} / h_{i}$ on $U_{i} \cap X$. To this end, first note that we can assume that the set $S$ is finite. This is because by the Nullstellensatz,
$$1 \in I(X)+\sum_{i \in S} I\left(\mathbb{A}{C}^{n} \backslash U{i}\right)$$
and so the indexing set $S$ can be made finite, since only finite sums of elements are allowed. Suppose, then,
$$S={1, \ldots, n}$$
Now also by the Nullstellensatz, there exist polynomials $a_{1}, \ldots, a_{n}, q$ such that $q \in I(X)$ and
$$a_{1} h_{1}+\cdots+a_{n} h_{n}+q=1$$
Then one verifies that the polynomial
$$\phi=a_{1} g_{1}+\cdots+a_{n} g_{n}$$
restricts to $g_{i} / h_{i}$ on each of the Zariski open sets $U_{i} \cap X$ (See Exercise 4.)

## 数学代考|代数几何代写algebraic geometry代考|Regular Functions on the Complement of a Set of Zeros

Let $X$ be an affine variety and let $f \in \mathbb{C}[X]$. Then the complement $X \backslash Z(f)$ is a special kind of quasiaffine variety. (We will see later that, in some sense, “it is still affine,” although not according to our current definition.) We have
$$\mathbb{C}[X \backslash Z(f)]=\mathbb{C}[X]\left[f^{-1}\right]=\mathbb{C}[X]\left[\frac{1}{f}\right]$$
Note that it is alright to take the reciprocal of $f$, because $X \backslash Z(f)$ does not contain any zeros of $f$. This construction is a special case of localization $S^{-1} R=R\left[S^{-1}\right]$ of a ring $R$ with respect to a subset $1 \in S \subseteq R$ closed under multiplication. On the set of “fractions” $r / s$ (i.e., formally, pairs $(r, s)$ ) with $r \in R$ and $s \in S$, define an equivalence relation where $r_{1} / s_{1} \sim r_{2} / s_{2}$ when there exists some $t \in S$ such that $r_{1} s_{2} t=r_{2} s_{1} t$. The set of equivalence classes is $R\left[S^{-1}\right]$. In our case, we let $S$ be the set of all $f^{n}, n \in \mathbb{N}_{0}$. We can alternately describe
$$R\left[f^{-1}\right]=R[t] /(f t-1)$$

The proof of (1.4.5) is actually essentially the same as in Sect. 1.4.3, with the exception that we have
$$a_{1} h_{1}+\cdots+a_{n} h_{n}=f^{N}$$
for some natural number $N$ (which is true by the Nullstellensatz). We then put
$$\phi=a_{1} g_{1} / f^{N}+\cdots+a_{n} g_{n} / f^{N}$$

## 数学代考|代数几何代写algebraic geometry代考|Regular Functions on a Quasiaffine Variety

Let $0 \neq f_{1}, \ldots, f_{m} \in \mathbb{C}[X]$ where $X$ is an affine variety. Then
$$\mathbb{C}\left[X \backslash Z\left(f_{1}, \ldots f_{m}\right)\right]=\mathbb{C}[X]\left[f_{1}^{-1}\right] \cap \cdots \cap \mathbb{C}[X]\left[f_{m}^{-1}\right]$$
Note that the intersection on the right hand side of (1.4.6) is formed in the field of rational functions $K(X)$, which is the field of fractions of the ring $\mathbb{C}[X]$. The field of fractions $Q R$ of an integral domain $R$ is the localization with respect to the set of all non-zero elements, which is closed under multiplication because $R$ is an integral domain. Also, the canonical map $R \rightarrow Q R$ is injective, by cancellation. Conversely, a subring of a field (more generally an integral domain) is obviously an integral domain. Thus, integral domains are precisely those rings which are subrings of fields.
The reason $\mathbb{C}[X]$ is an integral domain is that $X$ is irreducible. Now we have
$$X \backslash Z\left(f_{1}, \ldots, f_{m}\right)=\bigcup_{i=1}^{m} X \backslash Z\left(f_{i}\right)$$
Thus, we can characterize a regular function on $X \backslash Z\left(f_{1}, \ldots, f_{m}\right)$ by a collection of regular functions on $X \backslash Z\left(f_{i}\right)$ which coincide on intersections, but that is equivalent to coinciding in $K(X)$ since $\mathbb{C}[X]$, and hence also $\mathbb{C}[X]\left[f_{i}^{-1}\right]$, are integral domains. Thus, (1.4.6) follows.

## 数学代考|代数几何代写algebraic geometry代考|Regular Functions on an Affine Variety

p+一世(X)=p+q∣q∈一世(X)

1∈一世(X)+∑一世∈小号一世(一种Cn∖在一世)

φ=一种1G1+⋯+一种nGn

## 数学代考|代数几何代写algebraic geometry代考|Regular Functions on the Complement of a Set of Zeros

C[X∖从(F)]=C[X][F−1]=C[X][1F]

R[F−1]=R[吨]/(F吨−1)

(1.4.5) 的证明实际上与 Sect. 1.4.3，除了我们有

φ=一种1G1/Fñ+⋯+一种nGn/Fñ

## 数学代考|代数几何代写algebraic geometry代考|Regular Functions on a Quasiaffine Variety

C[X∖从(F1,…F米)]=C[X][F1−1]∩⋯∩C[X][F米−1]

X∖从(F1,…,F米)=⋃一世=1米X∖从(F一世)

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代考|代数几何代写algebraic geometry代考|Affine and Projective Varieties

statistics-lab™ 为您的留学生涯保驾护航 在代写代数几何algebraic geometry方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写代数几何algebraic geometry代写方面经验极为丰富，各种代写代数几何algebraic geometry相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代考|代数几何代写algebraic geometry代考|Projective and Quasi-Projective Varieties

The $n$-dimensional projective space $\mathbb{P}{\mathbb{C}}^{n}$ is the set of ratios $$\left[x{0}: \cdots: x_{n}\right]$$
of complex numbers. In a ratio, the numbers $x_{0}, \ldots, x_{n}$ are not allowed to all be 0 (although some may be 0 ), and a ratio is considered the same if we multiply all the numbers by the same non-zero number:
$$\left[x_{0}: \cdots: x_{n}\right]=\left[a x_{0}: \cdots: a x_{n}\right]$$
with $a \neq 0 \in \mathbb{C}$.
A projective algebraic set is a set of zeros in $\mathbb{P}{\mathbb{C}}^{n}$ of a set of homogeneous polynomials. (A polynomial is homogeneous if all its monomials have the same degree, which is defined as the sum of exponents of all its variables.) Projective algebraic sets are, by definition, the closed sets in the Zariski topology on $\mathbb{P}{\mathrm{C}}^{n}$. Irreducible projective algebraic sets are called projective varieties. A Zariski open subset (i.e. complement of a Zariski closed subset) in a projective variety is called a quasi-projective variety.One can, for many practical purposes, define an algebraic variety as a quasi-affine or quasi-projective variety. The necessity to always refer to an ambient affine or projective space in such a definition, however, is unsatisfactory, and it is a part of what motivates schemes. However, we must learn about varieties, and some other mathematics, first, before discussing schemes.

## 数学代考|代数几何代写algebraic geometry代考|Regular Functions on a Quasiaffine and Quasiprojective Variety

Let $V$ be a quasiaffine variety (or, more generally, a Zariski open set in an affine algebraic set). A regular function on $V$ at a point $p$ is a function
$$f: U \rightarrow \mathbb{C}$$
where $U$ is a Zariski open set in $V$ with $p \in U$ such that
$$f(x)=\frac{g(x)}{h(x)}$$
where $g(x), h(x)$ are polynomials, and $h(x) \neq 0$ for all $x \in U$ (here we write $x$ for an $n$-tuple: $\left.x=\left(x_{1}, \ldots, x_{n}\right)\right)$.
A regular function on $V$ is a function
$$f: V \rightarrow \mathbb{C}$$
which is regular at every point $p \in V$, i.e. for every $p \in V$, there exists a Zariski open neighborhood $U$ of $p$ such that on $U, f$ is of the form (1.4.1).

A regular function on a quasiprojective variety (or at a point of a quasiprojective variety) is defined the same way as a regular function on a quasiaffine variety with the exception that $g(x), h(x)$ are homogeneous polynomials of equal degree (so that $f(x)$ is well defined on ratios). The definition also applies to Zariski open subsets of projective algebraic sets.

Regular functions on an algebraic variety $X$ form a commutative ring (i.e. we can add and multiply them). This ring is denoted by $\mathbb{C}[X]$. We will now compute the ring of regular functions for varieties of certain kinds.

## 数学代考|代数几何代写algebraic geometry代考|Regular Functions on An

The ring of regular functions on the affine space is simply the ring of polynomials in $n$ variables:
$$\mathbb{C}\left[A_{C}^{n}\right]=\mathbb{C}\left[x_{1}, \ldots, x_{n}\right]$$
To see this, first note that since $A_{C}^{n}$ is irreducible, two polynomials $f, g$ which coincide on a non-empty Zariski open set $U \subseteq \AA_{\mathbb{C}}^{n}$ coincide (since $\mathrm{A}{\mathbb{C}}^{n}=\left(\mathrm{A}{\mathbb{C}}^{n} \backslash U\right) \cup Z(f-g)$ ). Now since $\mathrm{C}\left[x_{1}, \ldots, x_{n}\right]$ has unique factorization (see Theorem $4.1 .3$ below), the same is true for rational functions: Suppose on a non-empty Zariski open set $U \subseteq \mathbb{A}{\mathbb{C}}^{n}$, $$\frac{g{1}}{h_{1}}=\frac{g_{2}}{h_{2}}$$
where $g_{i}, h_{i}$ have greatest common divisor 1 for $i=1,2$, and $h_{i}$ are non-zero on $U$. Then
$$g_{1} h_{2}=g_{2} h_{1},$$
and hence there exists a $u \in \mathbb{C}^{\times}$such that $g_{1}=u g_{2}, h_{1}=u h_{2}$.
Now let $f$ be a regular function on $\mathbb{A}_{\mathbb{C}^{n}}^{n}$. But by what we just observed, in Zariski open neighborhoods of all points, we can write $f=g / h$ with the same polynomials $g, h$ which, moreover, have greatest common divisor 1 . However, if $h \notin \mathbb{C}^{\times}$, by the Nullstellensatz, then, the set of zeros $Z(h)$ of $h$ would be non-empty, so at a point $x \in Z(h)$, we would have a contradiction. Thus, $h \in \mathbb{C}^{\times}$, and $f$ is a polynomial.

## 数学代考|代数几何代写algebraic geometry代考|Projective and Quasi-Projective Varieties

\left[x_{0}: \cdots: x_{n}\right]=\left[a x_{0}: \cdots: a x_{n}\right]
$$with一种≠0∈C. 射影代数集是一组零磷Cn一组齐次多项式。（如果多项式的所有单项式都具有相同的次数，则多项式是齐次的，该次数被定义为所有变量的指数之和。）根据定义，射影代数集是 Zariski 拓扑中的闭集磷Cn. 不可约射影代数集称为射影簇。射影簇中的 Zariski 开子集（即 Zariski 闭子集的补集）称为准射影簇。出于许多实际目的，可以将代数簇定义为准仿射或准射影簇。然而，在这样的定义中总是指代环境仿射或投影空间的必要性是不能令人满意的，它是激发方案的一部分。但是，在讨论方案之前，我们必须首先了解变体和其他一些数学知识。 ## 数学代考|代数几何代写algebraic geometry代考|Regular Functions on a Quasiaffine and Quasiprojective Variety 让在是一个拟仿簇（或者，更一般地说，是仿射代数集中的一个 Zariski 开集）。一个常规函数在在某一点p是一个函数 F:在→C 在哪里在是一个 Zariski 开集在在和p∈在这样 F(X)=G(X)H(X) 在哪里G(X),H(X)是多项式，并且H(X)≠0对全部X∈在（这里我们写X为n-元组：X=(X1,…,Xn)). 一个常规函数在是一个函数 F:在→C 这在每一点都是正常的p∈在，即对于每个p∈在, 存在一个 Zariski 开放邻域在的p这样在在,F形式为 (1.4.1)。 一个准射影簇（或在一个准射影簇的一点）上的正则函数的定义方式与一个拟仿射簇上的正则函数相同，除了G(X),H(X)是等次的齐次多项式（所以F(X)在比率上有很好的定义）。该定义也适用于射影代数集的 Zariski 开子集。 代数簇上的正则函数X形成一个交换环（即我们可以将它们相加和相乘）。这个环表示为C[X]. 我们现在将计算某些种类的正则函数的环。 ## 数学代考|代数几何代写algebraic geometry代考|Regular Functions on An 仿射空间上的正则函数环就是多项式的环n变量： C[一种Cn]=C[X1,…,Xn] 要看到这一点，首先要注意，因为一种Cn是不可约的，两个多项式F,G在非空 Zariski 开集上重合在⊆\AACn巧合（因为一种Cn=(一种Cn∖在)∪从(F−G)）。现在自从C[X1,…,Xn]有唯一的因式分解（见定理4.1.3下面），对于有理函数也是如此：假设在一个非空的 Zariski 开集上在⊆一种Cn,G1H1=G2H2 在哪里G一世,H一世有最大公约数 1 为一世=1,2， 和H一世非零在. 然后 G1H2=G2H1, 因此存在一个在∈C×这样G1=在G2,H1=在H2. 现在让F成为一个常规函数一种Cnn. 但是根据我们刚刚观察到的，在所有点的 Zariski 开放邻域中，我们可以写F=G/H具有相同的多项式G,H此外，它的最大公约数为 1 。然而，如果H∉C×, 通过 Nullstellensatz, 那么, 零的集合从(H)的H将是非空的，所以在某一点上X∈从(H)，我们就会产生矛盾。因此，H∈C×， 和F是多项式。 统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。 ## 金融工程代写 金融工程是使用数学技术来解决金融问题。金融工程使用计算机科学、统计学、经济学和应用数学领域的工具和知识来解决当前的金融问题，以及设计新的和创新的金融产品。 ## 非参数统计代写 非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。 ## 广义线性模型代考 广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。 术语 广义线性模型（GLM）通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归，以及方差分析和方差分析（仅含固定效应）。 ## 有限元方法代写 有限元方法（FEM）是一种流行的方法，用于数值解决工程和数学建模中出现的微分方程。典型的问题领域包括结构分析、传热、流体流动、质量运输和电磁势等传统领域。 有限元是一种通用的数值方法，用于解决两个或三个空间变量的偏微分方程（即一些边界值问题）。为了解决一个问题，有限元将一个大系统细分为更小、更简单的部分，称为有限元。这是通过在空间维度上的特定空间离散化来实现的，它是通过构建对象的网格来实现的：用于求解的数值域，它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统，以模拟整个问题。然后，有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。 tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。 ## 随机分析代写 随机微积分是数学的一个分支，对随机过程进行操作。它允许为随机过程的积分定义一个关于随机过程的一致的积分理论。这个领域是由日本数学家伊藤清在第二次世界大战期间创建并开始的。 ## 时间序列分析代写 随机过程，是依赖于参数的一组随机变量的全体，参数通常是时间。 随机变量是随机现象的数量表现，其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值（如1秒，5分钟，12小时，7天，1年），因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中，往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录，以得到其自身发展的规律。 ## 回归分析代写 多元回归分析渐进（Multiple Regression Analysis Asymptotics）属于计量经济学领域，主要是一种数学上的统计分析方法，可以分析复杂情况下各影响因素的数学关系，在自然科学、社会和经济学等多个领域内应用广泛。 ## 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。 ## 数学代考|代数几何代写algebraic geometry代考|Zariski Topology 如果你也在 怎样代写代数几何algebraic geometry这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。 代数几何是数学的一个分支，经典地研究多变量多项式的零点。现代代数几何的基础是使用抽象代数技术，主要来自换元代数，以解决有关这些零点集的几何问题。 statistics-lab™ 为您的留学生涯保驾护航 在代写代数几何algebraic geometry方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写代数几何algebraic geometry代写方面经验极为丰富，各种代写代数几何algebraic geometry相关的作业也就用不着说。 我们提供的代数几何algebraic geometry及其相关学科的代写，服务范围广, 其中包括但不限于: • Statistical Inference 统计推断 • Statistical Computing 统计计算 • Advanced Probability Theory 高等概率论 • Advanced Mathematical Statistics 高等数理统计学 • (Generalized) Linear Models 广义线性模型 • Statistical Machine Learning 统计机器学习 • Longitudinal Data Analysis 纵向数据分析 • Foundations of Data Science 数据科学基础 ## 数学代考|代数几何代写algebraic geometry代考|Topology Algebraic geometry builds fundamental concepts of geometry out of pure algebra (rings and polynomials). A very basic concept of geometry is topology. A topology on a set X is specified by open (and/or closed) sets. An open set containing a point x \in X is also called an open neighborhood of x. A set X with a topology is called a topological space. An open set is the same thing as a complement of a closed set, and vice versa, so it suffices to specify either open sets or closed sets. Open sets in a topology are required to satisfy the following properties (or axioms): 1. \emptyset, X are open. 2. A union of arbitrarily (possibly infinitely) many open sets is open. 3. An intersection of two (hence finitely many) open sets is open. One can equivalently formulate the axioms for closed sets by swapping union and intersection. For any set S \subseteq X, we then have a smallest closed set (with respect to inclusion) \bar{S} containing S (namely, the intersection of all closed sets containing S ). It is called the closure of S. Symmetrically, the interior S^{\circ} is the largest open set (i.e. the union of all open sets) contained in S. ## 数学代考|代数几何代写algebraic geometry代考|Zariski and Analytic Topology In algebraic geometry, the set of all n-tuples of complex numbers is called the affine space \mathbb{A}{\mathrm{C}}^{n}. For the purposes of algebraic geometry, we consider the Zariski topology on \mathrm{A}{\mathrm{C}}^{n}, in which closed sets are affine algebraic sets (see Sect. 1.1). Similarly, in the Zariski topology on any affine algebraic set X, the closed sets are affine algebraic sets in A_{C}^{n} which are subsets of X. To verify the axioms of topology, one notes that for sets of n-variable polynomials S_{i}, we have$$
Z\left(\bigcup_{i} S_{i}\right)=\bigcap_{i} Z\left(S_{i}\right)
$$and for sets of n-variable polynomials S, T, we have$$
Z({p \cdot q \mid p \in S, q \in T})=Z(S) \cup Z(T)
$$The Zariski topology is not the most typical kind of topology one considers outside of algebraic geometry. In analysis, the key example of a topology is the analytic topology. In the analytic topology on \mathbb{A}{C}^{n}=\mathbb{C}^{n} (or on \mathbb{R}^{n} ), a set U is open when with any point x \in U, the set U also contains all points of distance <\epsilon for some \epsilon>0 (where \epsilon can depend on x ). As the name suggests, the analytic topology is very important in mathematical analysis. The Zariski topology has “far fewer” closed (and open) sets than the analytic topology. For example, in \mathbb{R}^{n} (or \mathbb{C}^{n} ), any open ball is open and any closed ball is closed in the analytic topology. On the other hand, the only Zariski closed sets in A{C}^{1} are itself and finite subsets. Still, we can use the analytic topology for intuition about the Zariski topology on algebraic sets. For example, a single point is closed (in both analytic and Zariski topology), and is not open, unless we are in \mathrm{A}_{\mathrm{C}^{*}}^{0}. ## 数学代考|代数几何代写algebraic geometry代考|Affine and Quasi-Affine Varieties In a topological space X, a non-empty closed set Z is called irreducible if there do not exist closed subsets Z_{1} \neq Z, Z_{2} \neq Z of Z such that Z=Z_{1} \cup Z_{2} (i.e. Z is not a union of two closed subsets other than itself). Z is called connected if it is not a union of two disjoint closed subsets other than itself. An affine variety is an affine algebraic set which is irreducible in the Zariski topology. A quasi-affine variety is a Zariski open subset U of an affine variety X. (Caution: U is open a topology on S where, by definition, open (resp. closed) sets in S are of the form V \cap S where V is an open (resp. closed) set in X. This topology is called the induced topology. The Zariski topology on an affine algebraic set is induced from the Zariski topology on \mathrm{A}^{n} \mathrm{C}. Recall that an ideal I \subseteq R in a ring R is called prime if I \neq R and for x, y \in R, x y \in I implies x \in I or y \in I. The ideal I is called maximal if I \neq R and for every ideal J \subseteq R with I \subseteq J, we have J=I or J=R. An ideal I \subseteq R is maximal if and only if the quotient ring R / I (consisting of all cosets x+I, x \in R ) is a field. Similarly, I \subseteq R is prime if and only if R / I is an integral domain which means that it satisfies 0 \neq 1 and has no zero divisors (i.e. non-zero elements x, y such that x y=0 ). Any ideal I \neq R is contained in a maximal ideal by a principle called Zorn’s lemma, which states that any partially ordered set P (such as the set of ideals in a ring R ordered with respect to inclusion) contains a maximal element (i.e. an element m \in P such that a \in P and m \leq a implies m=a ), provided that for any subset L which is totally ordered (i.e. a, b \in L implies a \leq b or b \leq a ) there exists an element \ell \in P greater or equal than all elements of L. Now it is easy to see that an affine algebraic set X is irreducible (i.e. is an affine variety) if and only if the ideal I(X) is prime. Indeed, if I(X) is not prime, then there exists f, g \notin I(X) such that f g \in I(X), so X is a union of the two closed subsets Z(f) \cap X, Z(g) \cap X neither of which is equal to X. On the other hand, if X=X_{1} \cup X_{2} where X_{i} \neq X are closed, then by definition, there are f_{i} \in I\left(X_{i}\right) \backslash I(X), while f_{1} f_{2} \in I(X). In particular, since polynomials over a field obviously form an integral domain, the 0 ideal is prime, and thus, the affine space \mathrm{A}_{C}^{n} is irreducible (and hence, an affine variety). ## 代数几何代写 ## 数学代考|代数几何代写algebraic geometry代考|Topology 代数几何从纯代数（环和多项式）中构建了几何的基本概念。几何的一个非常基本的概念是拓扑。集合上的拓扑X由开放（和/或封闭）集指定。包含一个点的开集X∈X也称为开放邻域X. 一套X有拓扑的称为拓扑空间。开集与闭集的补集相同，反之亦然，因此指定开集或闭集就足够了。拓扑中的开集需要满足以下属性（或公理）： 1. ∅,X是开放的。 2. 任意（可能无限）许多开集的并集是开集。 3. 两个（因此是有限多个）开集的交集是开集。 可以通过交换并集和交集来等效地制定封闭集的公理。 对于任何集合小号⊆X，然后我们有一个最小的闭集（关于包含）小号¯包含小号（即所有包含的闭集的交集小号）。它被称为闭包小号. 对称地，内部小号∘是最大的开集（即所有开集的并集）包含在小号. ## 数学代考|代数几何代写algebraic geometry代考|Zariski and Analytic Topology 在代数几何中，所有的集合n-复数的元组称为仿射空间一种Cn. 为了代数几何的目的，我们考虑 Zariski 拓扑一种Cn，其中闭集是仿射代数集（参见第 1.1 节）。类似地，在任何仿射代数集上的 Zariski 拓扑中X, 闭集是仿射代数集一种Cn是的子集X. 为了验证拓扑公理，人们注意到对于n- 变量多项式小号一世， 我们有 从(⋃一世小号一世)=⋂一世从(小号一世) 并且对于n- 变量多项式小号,吨， 我们有 从(p⋅q∣p∈小号,q∈吨)=从(小号)∪从(吨) Zariski 拓扑不是代数几何之外最典型的拓扑。在分析中，拓扑的关键示例是解析拓扑。在解析拓扑上一种Cn=Cn（或在Rn）， 一套在任何时候都打开X∈在, 集合在还包含所有距离点<ε对于一些ε>0（在哪里ε可以依赖X）。顾名思义，解析拓扑在数学分析中非常重要。Zariski 拓扑的封闭（和开放）集比解析拓扑“少得多”。例如，在Rn（或者Cn)，在解析拓扑中，任何开球都是开球，任何闭球都是闭球。另一方面，Zariski 唯一的封闭式一种C1是自身和有限子集。 尽管如此，我们仍然可以使用解析拓扑来直观了解代数集上的 Zariski 拓扑。例如，一个点是封闭的（在解析拓扑和 Zariski 拓扑中），并且不是开放的，除非我们在一种C∗0. ## 数学代考|代数几何代写algebraic geometry代考|Affine and Quasi-Affine Varieties 在拓扑空间中X, 一个非空闭集从如果不存在封闭子集，则称为不可约从1≠从,从2≠从的从这样从=从1∪从2（IE从不是两个封闭子集的并集，而不是自身）。从如果它不是除自身之外的两个不相交的封闭子集的并集，则称为连通的。 仿射簇是在 Zariski 拓扑中不可约的仿射代数集。准仿射变体是 Zariski 开子集在仿射品种X. （警告：在打开拓扑小号其中，根据定义，打开（或关闭）设置在小号是形式在∩小号在哪里在是一个开（或闭）集X. 这种拓扑称为诱导拓扑。 仿射代数集上的 Zariski 拓扑是从上的 Zariski 拓扑推导出来的一种nC. 回想一下理想一世⊆R在一圈R称为素数，如果一世≠R并且对于X,是∈R, X是∈一世暗示X∈一世或者是∈一世. 理想一世被称为最大如果一世≠R对于每一个理想Ĵ⊆R和一世⊆Ĵ， 我们有Ĵ=一世或者Ĵ=R. 一个理想一世⊆R最大当且仅当商环R/一世（由所有陪衬组成X+一世,X∈R) 是一个字段。相似地，一世⊆R是素数当且仅当R/一世是一个积分域，这意味着它满足0≠1并且没有零除数（即非零元素X,是这样X是=0 ). 任何理想一世≠R被称为 Zorn 引理的原理包含在极大理想中，该原理表明任何偏序集磷（例如环中的一组理想R相对于包含排序）包含一个最大元素（即一个元素米∈磷这样一种∈磷和米≤一种暗示米=一种)，前提是对于任何子集大号这是完全有序的（即一种,b∈大号暗示一种≤b或者b≤一种) 存在一个元素ℓ∈磷大于或等于的所有元素大号. 现在很容易看出仿射代数集X是不可约的（即是仿射变体）当且仅当理想一世(X)是素数。确实，如果一世(X)不是素数，则存在F,G∉一世(X)这样FG∈一世(X)， 所以X是两个封闭子集的并集从(F)∩X, 从(G)∩X两者都不等于X. 另一方面，如果X=X1∪X2在哪里X一世≠X是封闭的，那么根据定义，有F一世∈一世(X一世)∖一世(X)， 尽管F1F2∈一世(X). 特别是，由于域上的多项式显然形成了一个整数域，所以 0 理想是素数，因此，仿射空间一种Cn是不可约的（因此是仿射变体）。 统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。 ## 金融工程代写 金融工程是使用数学技术来解决金融问题。金融工程使用计算机科学、统计学、经济学和应用数学领域的工具和知识来解决当前的金融问题，以及设计新的和创新的金融产品。 ## 非参数统计代写 非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。 ## 广义线性模型代考 广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。 术语 广义线性模型（GLM）通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归，以及方差分析和方差分析（仅含固定效应）。 ## 有限元方法代写 有限元方法（FEM）是一种流行的方法，用于数值解决工程和数学建模中出现的微分方程。典型的问题领域包括结构分析、传热、流体流动、质量运输和电磁势等传统领域。 有限元是一种通用的数值方法，用于解决两个或三个空间变量的偏微分方程（即一些边界值问题）。为了解决一个问题，有限元将一个大系统细分为更小、更简单的部分，称为有限元。这是通过在空间维度上的特定空间离散化来实现的，它是通过构建对象的网格来实现的：用于求解的数值域，它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统，以模拟整个问题。然后，有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。 tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。 ## 随机分析代写 随机微积分是数学的一个分支，对随机过程进行操作。它允许为随机过程的积分定义一个关于随机过程的一致的积分理论。这个领域是由日本数学家伊藤清在第二次世界大战期间创建并开始的。 ## 时间序列分析代写 随机过程，是依赖于参数的一组随机变量的全体，参数通常是时间。 随机变量是随机现象的数量表现，其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值（如1秒，5分钟，12小时，7天，1年），因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中，往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录，以得到其自身发展的规律。 ## 回归分析代写 多元回归分析渐进（Multiple Regression Analysis Asymptotics）属于计量经济学领域，主要是一种数学上的统计分析方法，可以分析复杂情况下各影响因素的数学关系，在自然科学、社会和经济学等多个领域内应用广泛。 ## 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。 ## 数学代考|代数几何代写algebraic geometry代考|The Definition of Algebraic Varieties 如果你也在 怎样代写代数几何algebraic geometry这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。 代数几何是数学的一个分支，经典地研究多变量多项式的零点。现代代数几何的基础是使用抽象代数技术，主要来自换元代数，以解决有关这些零点集的几何问题。 statistics-lab™ 为您的留学生涯保驾护航 在代写代数几何algebraic geometry方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写代数几何algebraic geometry代写方面经验极为丰富，各种代写代数几何algebraic geometry相关的作业也就用不着说。 我们提供的代数几何algebraic geometry及其相关学科的代写，服务范围广, 其中包括但不限于: • Statistical Inference 统计推断 • Statistical Computing 统计计算 • Advanced Probability Theory 高等概率论 • Advanced Mathematical Statistics 高等数理统计学 • (Generalized) Linear Models 广义线性模型 • Statistical Machine Learning 统计机器学习 • Longitudinal Data Analysis 纵向数据分析 • Foundations of Data Science 数据科学基础 ## 数学代考|代数几何代写algebraic geometry代考|Affine Algebraic Sets The starting point of algebraic geometry is studying solutions of systems of polynomial equations in several variables over a field. (A field is an algebraic structure with operations of addition and multiplication which satisfy all the formal properties of the real numbers, i.e. commutativity and associativity of both operations, the existence of 0 \neq 1 with their usual properties, distributivity, and the existence of an additive inverse-or minus signas well as multiplicative inverses of non-zero elements). Systems of polynomial equations in variables x_{1}, \ldots, x_{n} can always be written in the form$$
\begin{array}{r}
p_{1}\left(x_{1}, \ldots, x_{n}\right)=0 \
\ldots \
p_{m}\left(x_{1}, \ldots, x_{n}\right)=0
\end{array}
$$where p_{1}, \ldots, p_{m} are polynomials. Solutions of the Eqs. (1.1.1) are n-tuples of elements \left(x_{1}, \ldots, x_{n}\right) of the given field which satisfy the equations. Such n-tuples are also called zeros of the polynomials p_{1}, \ldots, p_{n}. Sets of zeros of sets of polynomials are called affine algebraic sets. The set of all polynomials in n variables over a field forms a commutative ring, which means that it has operations of addition and multiplication satisfying all the formal properties of integers, i.e. commutativity and associativity of both operations, the existence of 0 and 1 , distributivity, and the existence of an additive inverse. One defines not necessarily commutative rings by dropping the assumption that multiplication be commutative (we then must require that 1 be a left and right unit and that left and right distributivities hold). In this book, by a ring, we shall mean a commutative ring, unless specified otherwise. Solutions of (1.1.1), or zeros of the polynomials p_{1}, \ldots, p_{m}, are also zeros of all linear combinations$$
a_{1} p_{1}+\cdots+a_{m} p_{m}
$$where a_{1}, \ldots, a_{m} are arbitrary polynomials. The elements (1.1.2) form the ideal generated by p_{1}, \ldots, p_{m}, which is denoted by$$
\left(p_{1}, \ldots, p_{m}\right)
$$An ideal in a commutative ring is a subset which contains 0 , is closed under +, and multiples by elements of the ring. By the Hilbert basis theorem, which we prove in Sect. 4 (Theorem 4.2.1), the ring of polynomials in n variables over a field is Noetherian, which means that every ideal is finitely generated (i.e. generated by finitely many elements). Because of this, it is sufficient to consider systems of finitely many polynomial equations (1.1.1). Note that a commutative ring R is Noetherian if and only if it satisfies the ascending chain condition (ACC) with respect to ideals. To satisfy the ACC with respect to subsets of a certain kind means that there does not exist an infinite chain$$
I_{1} \subsetneq I_{2} \subsetneq \cdots \subsetneq I_{n} \subsetneq \cdots
$$of such sets. Thus, we claim that a ring R is Noetherian if and only if (1.1.3) does not occur in R where I_{n} are ideals. To see this, if R is not Noetherian, it has an ideal I which is not finitely generated, so having picked, by induction, elements r_{1} \ldots, r_{n} \in I, they cannot generate I, so we can pick r_{n+1} \in I \backslash\left(r_{1}, \ldots, r_{n}\right). Thus, R fails the ACC for ideals. On the other hand, if R fails the ACC for ideals, then we have ideals (1.1.3) in R. Assume, for contradiction, that R is Noetherian. Let$$
I=\bigcup_{n} I_{n} .
$$Then the ideal is finitely generated, say, by elements r_{1}, \ldots, r_{k}. Thus, there exists an n such that r_{1}, \ldots, r_{k} \in I_{n}, which implies I_{n}=I, which is a contradiction. ## 数学代考|代数几何代写algebraic geometry代考|Complex Numbers Zeros of polynomials behave better when we work in the field \mathbb{C} of complex numbers than in the field \mathbb{R} of real numbers. The field \mathbb{C} contains the number i which has the property$$
i^{2}=-1
$$and more generally, a complex number can be uniquely written as a+b i where a, b are real numbers. Addition and multiplication are then determined by the properties of a field. Division is possible because (a+b i)(a-b i)=a^{2}+b^{2}, and we can thus make the denominator real. Thus, the polynomial equation$$
x^{2}+1=0
$$has solutions in \mathbb{C}, namely i and -i, while it has no solution over the field of real numbers \mathbb{R}. It turns out that more generally, every non-constant polynomial in one variable with coefficients in \mathbb{C} has at least one zero (we also say root). A field which satisfies this property is called algebraically closed. The fact that \mathbb{C} is algebraically closed is known as the fundamental theorem of algebra. In the first three sections of this chapter, we will assume from now on that we are working over the field \mathbb{C}. More generally, in much of what we say (excluding connections with analysis), we could work over any algebraically closed field. ## 数学代考|代数几何代写algebraic geometry代考|Nullstellensatz The fact that a non-constant polynomial over \mathbb{C} always has a root can be generalized to several variables as follows: Let I be an ideal in the ring \mathbb{C}\left[x_{1}, \ldots, x_{n}\right] of polynomials in n variables over \mathbb{C}. Let X=Z(I) be the affine algebraic set which is the set of zeros of the ideal I. If I=\left(f_{1}, \ldots, f_{m}\right), we also write$$
Z\left(f_{1}, \ldots, f_{m}\right)=Z(I)
$$Let, on the other hand, I(X) be the ideal of all polynomials which are zero on X (i.e. p\left(x_{1}, \ldots, x_{n}\right)=0 for every \left.\left(x_{1}, \ldots, x_{n}\right) \in X\right). Then$$
I(X)=\sqrt{I}

where the right hand side of (1.1.4) is called the radical of $I$ and consists of all polynomials $p$ for which $p^{k} \in I$ for some non-negative integer $k$. Equation (1.1.4) is called the Nullstellensatz, and is due to Hilbert. In German, Nullstelle means zero, literally “zero place,” a point at which a polynomial is zero. In English, as we already remarked, such a point is called just a “zero,” which can be confusing.

As many facts in algebraic geometry, a proof of the Nullstellensatz requires certain methods from algebra. The kind of algebra relevant to the foundations of algebraic geometry is known as commutative algebra, to which we will keep returning throughout this book. The Nullstellensatz will be restated and proved in Sect. $4.3$ below.

## 数学代考|代数几何代写algebraic geometry代考|Affine Algebraic Sets

p1(X1,…,Xn)=0 … p米(X1,…,Xn)=0

(1.1.1) 的解，或多项式的零点p1,…,p米, 也是所有线性组合的零

(p1,…,p米)

X2+1=0

## 有限元方法代写

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## 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。