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

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

An algebraic identity is an equality between two elements of $\mathbb{Z}\left[X_1, \ldots, X_n\right]$ defined differently. It gets automatically transferred into every commutative ring by means of the previous corollary.

Since the ring $\mathbb{Z}\left[X_1, \ldots, X_n\right]$ has particular properties, it happens that some algebraic identities are easier to prove in $\mathbb{Z}\left[X_1, \ldots, X_n\right]$ than in “an arbitrary ring $\mathbf{B}$.” Consequently, if the structure of a theorem reduces to a family of algebraic identities, which is very frequent in commutative algebra, it is often in our interest to use a ring of polynomials with coefficients in $\mathbb{Z}$ by taking as its indeterminates the relevant elements in the statement of the theorem.

The properties of the rings $\mathbb{Z}[\underline{X}]$ which may prove useful are numerous. The first is that it is an integral ring. So it is a subring of its quotient field $\mathbb{Q}\left(X_1, \ldots, X_n\right)$ which offers all the facilities of discrete fields.

The second is that it is an infinite and integral ring. Consequently, “all bothersome but rare cases can be ignored.” A case is rare when it corresponds to the annihilation of a polynomial $Q$ that evaluates to zero everywhere. It suffices to check the equality corresponding to the algebraic identity when it is evaluated at the points of $\mathbb{Z}^n$ which do not annihilate $Q$. Indeed, if the algebraic identity we need to prove is $P=0$, we get that the polynomial $P Q$ defines the function over $\mathbb{Z}^n$ that evaluates to zero everywhere, this implies that $P Q=0$ and thus $P=0$ since $Q \neq 0$ and $\mathbb{Z}[\underline{X}]$ is integral. This is sometimes called the “extension principle for algebraic identities.”
Other remarkable properties of $\mathbb{Z}[\underline{X}]$ could sometimes be used, like the fact that it is a unique factorization domain (UFD) as well as being a strongly discrete coherent Noetherian ring of finite Krull dimension.

## 数学代写|交换代数代写Commutative Algebra代考|Weights, Homogeneous Polynomials

We say that we have defined a weight on a polynomial algebra $\mathbf{A}\left[X_1, \ldots, X_k\right]$ when we attribute to each indeterminate $X_i$ a weight $w\left(X_i\right) \in \mathbb{N}$. We then define the weight of the monomial $\underline{X} \underline{\underline{m}}=X_1^{m_1} \cdots X_k^{m_k}$ as
$$w\left(\underline{X}^{\underline{m}}\right)=\sum_i m_i w\left(X_i\right)$$
so that $w\left(\underline{X}^{\underline{m}}+\underline{m^{\prime}}\right)=w\left(\underline{X}^{\underline{m}}\right)+w\left(\underline{X}^{m^{\prime}}\right)$. The degree of a polynomial $P$ for this weight, generally denoted by $w(P)$, is the greatest of the weights of the monomials appearing with a nonzero coefficient. This is only well-defined if we have a test of equality to 0 in $\mathbf{A}$ at our disposal. In the opposite case we simply define the statement ” $w(P) \leqslant r . “$

A polynomial is said to be homogeneous (for a weight $w$ ) if all of its monomials have the same weight.

When we have an algebraic identity and a weight available, each homogeneous component of the algebraic identity provides a particular algebraic identity.

We can also define weights with values in some monoids with a more complicated order than $(\mathbb{N}, 0,+, \geqslant)$. We then ask that this monoid be the positive part of a product of totally ordered Abelian groups, or more generally a monoid with gcd (this notion will be introduced in Chap. XI).

Symmetric Polynomials
We fix $n$ and $\mathbf{A}$ and we let $S_1, \ldots, S_n$ be the elementary symmetric polynomials at the $X_i$ ‘s in $\mathbf{A}\left[X_1, \ldots, X_n\right]$. They are defined by the equality
$$T^n+S_1 T^{n-1}+S_2 T^{n-2}+\cdots+S_n=\prod_{i=1}^n\left(T+X_i\right) .$$
We have $S_1=\sum_i X_i, S_n=\prod_i X_i, S_k=\sum_{J \in \mathcal{P}{k, n}} \prod{i \in J} X_i$. Recall the following well-known theorem (a proof is suggested in Exercise 3).

## 数学代写|交换代数代写Commutative Algebra代考|Weights, Homogeneous Polynomials

$$w\left(\underline{X}^{\underline{m}}\right)=\sum_i m_i w\left(X_i\right)$$

$$T^n+S_1 T^{n-1}+S_2 T^{n-2}+\cdots+S_n=\prod_{i=1}^n\left(T+X_i\right) .$$

## 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。