Posted on Categories:Algebraic Geometry, 代数几何, 数学代写

数学代写|代数几何代写Algebraic Geometry代考|MATH413 Applications

avatest™

avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

数学代写|代数几何代写Algebraic Geometry代考|Applications

Unfortunately, computing over algebraically closed fields presents significant technical difficulties. How can a computer represent a general complex number? The Hilbert Nullstellensatz gives us a procedure for deciding whether a polynomial vanishes over the complex points of variety without ever computing over the complex numbers! We just need to check whether the polynomial is contained in the radical of the ideal generated by some defining set of equations. This can be checked over any field containing the coefficients of the polynomials at hand.

We no longer assume that $k$ is algebraically closed. To test whether a polynomial $g \in \sqrt{I}$, where $I=\left\langle f_1, \ldots, f_r\right\rangle$, we use the following criterion.

Proposition7.21 Given an ideal $I=\left\langle f_1, \ldots, f_r\right\rangle \subset k\left[x_1, \ldots, x_n\right], g \in \sqrt{I}$ if and only if $\left\langle f_1, \ldots, f_r, z g-1\right\rangle=k\left[x_1, \ldots, x_n, z\right]$.
Proof The proof of the Hilbert Nullstellensatz gives that
$1 \in\left\langle f_1, \ldots, f_r, z g-1\right\rangle \Rightarrow g^N \in I$ for some $N$.

Conversely, if $g^N=f_1 h_1+\cdots+f_r h_r$ then $z^N g^N=f_1\left(h_1 z^N\right)+\cdots+f_r\left(h_r z^N\right)$ and
$$1=f_1\left(h_1 z^N\right)+\cdots+f_r\left(h_r z^N\right)+\left(1-z^N g^N\right) .$$
Since $\left(1-z^N g^N\right)=(1-z g)\left(1+z g+\cdots+z^{N-1} g^{N-1}\right)$, we conclude $1 \in\left\langle f_1, \ldots\right.$, $\left.f_r, z g-1\right\rangle$.

数学代写|代数几何代写Algebraic Geometry代考|Dimension

Definition 7.26 Let $V \subset \mathbb{A}^n(k)$ be an irreducible affine variety. The dimension $\operatorname{dim} V$ is defined as the transcendence degree of $k(V)$ over $k$.

We outline an effective procedure to compute the dimension of a variety. Let $I \subset k\left[x_1, \ldots, x_n\right]$ be a prime ideal, $F$ the quotient field of $k\left[x_1, \ldots, x_n\right] / I$, and $d$ the transcendence degree of $F$ over $k$.
By Proposition $7.15$, there exist indices
$$1 \leq i_1<i_2<\ldots<i_d \leq n$$
such that $x_{i_1}, \ldots, x_{i_d}$ form a trascendence basis of $F$ over $k$. Indeed, any maximal algebraically independent subset will do. We therefore focus on determining whether a subset of the variables is algebraically independent. For notational simplicity, we take the first few variables.

Proposition 7.27 The elements $x_1, \ldots, x_e \in F$ are algebraically independent over $k$ if and only if $I \cap k\left[x_1, \ldots, x_e\right]=0$.

The intersection can be effectively computed using the Elimination Theorem (Theorem 4.8)

Proof If $x_1, \ldots, x_e$ are algebraically dependent then there exists a nonzero polynomial $f \in k\left[t_1, \ldots, t_e\right]$ such that $f\left(x_1, \ldots, x_e\right) \equiv 0(\bmod I)$. This gives a nontrivial element of $I \cap k\left[x_1, \ldots, x_e\right]$. Conversely, each such element gives an algebraic dependence relation among $x_1, \ldots, x_e$.

数学代写|代数几何代写Algebraic Geometry代考|Applications

$1 \in\left\langle f_1, \ldots, f_r, z g-1\right\rangle \Rightarrow g^N \in I$ 对于一些 $N$.

$$1=f_1\left(h_1 z^N\right)+\cdots+f_r\left(h_r z^N\right)+\left(1-z^N g^N\right) .$$

数学代写|代数几何代写Algebraic Geometry代考|Dimension

$$1 \leq i_1<i_2<\ldots<i_d \leq n$$

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。