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数学代写|代数几何代写Algebraic Geometry代考|Gro¨bner bases and the division algorithm

Algorithm $2.7$ (Division procedure) $\quad$ Fix a monomial order $>$ on $k\left[x_1, \ldots, x_n\right]$ and nonzero polynomials $f_1, \ldots, f_r \in k\left[x_1, \ldots, x_n\right]$. Given $g \in k\left[x_1, \ldots, x_n\right]$, we want to determine whether $g \in\left\langle f_1, \ldots, f_r\right\rangle$ :
Step o Put $g_0=g$. If there exists no $f_j$ with $\operatorname{LM}\left(f_j\right) \mid L M\left(g_0\right)$ then we STOP. Otherwise, pick such an $f_{j_0}$ and cancel leading terms by putting
$$g_1=g_0-f_{j_0} \mathrm{LT}\left(g_0\right) / \mathrm{LT}\left(f_{j_0}\right) .$$
Step i Given $g_i$, if there exists no $f_j$ with $\operatorname{LM}\left(f_j\right) \mid \mathrm{LM}\left(g_i\right)$ then we STOP. Otherwise, pick such an $f_{j_i}$ and cancel leading terms by putting
$$g_{i+1}=g_i-f_{j_i} \operatorname{LT}\left(g_i\right) / \operatorname{LT}\left(f_{j_i}\right) .$$
As we are cancelling leading terms at each stage, we have
$$\mathrm{LM}(g)=\mathrm{LM}\left(g_0\right)>\mathrm{LM}\left(g_1\right)>\ldots>\operatorname{LM}\left(g_i\right)>\mathrm{LM}\left(g_{i+1}\right)>\ldots$$

By the well-ordering property of the monomial order, such a chain of decreasing monomials must eventually terminate. If this procedure does not stop, then we must have $g_N=0$ for some $N$. Back-substituting using Equation 2.1, we obtain
$$g=\sum_{i=0}^{N-1} f_{j_i} \operatorname{LT}\left(g_i\right) / \operatorname{LT}\left(f_{j_i}\right)=\sum_{j=1}^r\left(\sum_{j_i=j} \operatorname{LT}\left(g_i\right) / \operatorname{LT}\left(f_{j_i}\right)\right) f_j=\sum_{j=1}^r h_j f_j,$$
where the last sum is obtained by regrouping terms.
Unfortunately, this procedure often stops prematurely. Even when $g \in$ $\left\langle f_1, \ldots, f_r\right\rangle$, it may happen that $\mathrm{LM}(g)$ is not divisible by any $\operatorname{LM}\left(f_j\right)$.

数学代写|代数几何代写Algebraic Geometry代考|Normal forms

Theorem 2.16 Fix a monomial order $>$ on $k\left[x_1, \ldots, x_n\right]$ and an ideal $I \subset$ $k\left[x_1, \ldots, x_n\right]$. Then each $g \in k\left[x_1, \ldots, x_n\right]$ has a unique expression
$$g \equiv \sum_{x^\alpha \notin \operatorname{LT}(I)} c_\alpha x^\alpha(\bmod I),$$
where $c_\alpha \in k$ and all but a finite number are zero. The expression $\sum_\alpha c_\alpha x^\alpha$ is called the normal form of $g$ modulo $I$.

Equivalently, the monomials $\left{x^\alpha: x^\alpha \notin \operatorname{LT}(I)\right}$ form a $k$-vector-space basis for the quotient $k\left[x_1, \ldots, x_n\right] / I$.

Corollary 2.17 Fix a monomial order $>$ on $k\left[x_1, \ldots, x_n\right]$, an ideal $I \subset$ $k\left[x_1, \ldots, x_n\right]$, and Gröbner basis $f_1, \ldots, f_r$ for I. Then each $g \in k\left[x_1, \ldots, x_n\right]$ has a unique expression
$$g \equiv \sum c_\alpha x^\alpha \quad(\bmod I),$$
where $\mathrm{LM}\left(f_j\right)$ does not divide $x^\alpha$ for any $j$ or $\alpha$.
Proof of theorem: We first establish existence: the proof is essentially an induction on $\operatorname{LM}(g)$. Suppose the result is false, and consider the nonempty set
${\mathrm{LM}(g): g$ does not admit a normal form $}$
One of the defining properties of monomial orders guarantees that this set has a least element $x^\beta$; choose $g$ such that $\operatorname{LT}(g)=x^\beta$.

数学代写|代数几何代写Algebraic Geometry代考|Gro”bner bases and the division algorithm

$$g_1=g_0-f_{j_0} \mathrm{LT}\left(g_0\right) / \mathrm{LT}\left(f_{j_0}\right) .$$

$$g_{i+1}=g_i-f_{j_i} \mathrm{LT}\left(g_i\right) / \operatorname{LT}\left(f_{j_i}\right) .$$

$$\operatorname{LM}(g)=\operatorname{LM}\left(g_0\right)>\operatorname{LM}\left(g_1\right)>\ldots>\operatorname{LM}\left(g_i\right)>\operatorname{LM}\left(g_{i+1}\right)>\ldots$$

$$g=\sum_{i=0}^{N-1} f_{j_i} \mathrm{LT}\left(g_i\right) / \mathrm{LT}\left(f_{j_i}\right)=\sum_{j=1}^r\left(\sum_{j_i=j} \mathrm{LT}\left(g_i\right) / \mathrm{LT}\left(f_{j_i}\right)\right) f_j=\sum_{j=1}^r h_j f_j$$

数学代写|代数几何代写Algebraic Geometry代考|Normal forms

$$g \equiv \sum_{x^\alpha \nless \operatorname{LT}(I)} c_\alpha x^\alpha(\bmod I)$$

$$g \equiv \sum c_\alpha x^\alpha \quad(\bmod I)$$

$\mathrm{LM}(g): g \$$doesnotadmitanormal form \$$ 单项式的定义属性之一保证该集合具有最小元筙$x^\beta$; 选择$g$这样$\mathrm{LT}(g)=x^\beta\$.

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。