# 线性代数网课代修|交换代数代写Commutative Algebra代考|MATH4312

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• 数值分析
• 高等线性代数
• 矩阵论
• 优化理论
• 线性规划
• 逼近论

## 线性代数作业代写linear algebra代考|The Implicitization Problem

Example 5. Now consider the tangent surface of the twisted cubic in $\mathbb{R}^{3}$, which we studied in Chapter 1 . This surface is parametrized by
\begin{aligned} &x=t+u, \ &y=t^{2}+2 t u, \ &z=t^{3}+3 t^{2} u \end{aligned}
We compute a Groebner basis $G$ for this ideal relative to the lex order defined by $t>u>x>y>z$, and we find that $G$ has 6 elements altogether. If you make the calculation, you will see that only one contains only $x, y, z$ terms:
$$-(4 / 3) x^{3} z+x^{2} y^{2}+2 x y z-(4 / 3) y^{3}-(1 / 3) z^{2}=0 .$$
The variety defined by this equation is a surface containing the tangent surface to the twisted cubic. However, it is possible that the surface given by (4) is strictly bigger than the tangent surface: there may be solutions of (4) that do not correspond to points on the tangent surface. We will return to this example in Chapter $3 .$

To summarize our findings in this section, we have seen that Groebner bases and the division algorithm give a complete solution of the ideal membership problem. Furthermore, we have seen ways to produce solutions of systems of polynomial equations and to produce equations of parametrically given subsets of affine space. Our success in the examples given earlier depended on the fact that Groebner bases, when computed using lex order, seem to eliminate variables in a very nice fashion. In Chapter 3 , we will prove that this is always the case, and we will explore other aspects of what is called elimination theory.

## 线性代数作业代写linear algebra代考|Improvements on Buchberger’s Algorithm

In designing useful mathematical software, attention must be paid not only to the correctness of the algorithms employed, but also to their efficiency. In this section, we will discuss some improvements on the basic Buchberger algorithm for computing Groebner bases that can greatly speed up the calculations. Some version of these improvements has been built into most of the computer algebbra systems that offer Groebner basis packages. The section will conclude with a brief discussion of the complexity of Buchberger’s algorithm. This is still an active area of research though, and there are as yet no definitive results in this direction.

The first class of modifications we will consider concern Theorem 6 of $\S 6$, which states that an ideal basis $G$ is a Groebner basis provided that $\overline{S(f, g)}^{G}=0$ for all $f, g \in G$. If you look back at $\S 7$, you will see that this criterion is the driving force behind Buchberger’s algorithm. Hence, a good way to improve the efficiency of the algorithm would be to show that fewer S-polynomials $S(f, g)$ need to be considered. As you learned from doing examples by hand, the polynomial divisions involved are the most computationally intensive part of Buchberger’s algorithm. Thus, any reduction of the number of divisions that need to be performed is all to the good.

To identify $S$-polynomials that can be ignored in Theorem 6 of $\S 6$, we first need to give a more general view of what it means to have zero remainder. The definition is as follows.

Definition 1. Fix a monomial order and let $G=\left{g_{1}, \ldots, g_{s}\right} \subset k\left[x_{1}, \ldots, x_{n}\right]$. Given $f \in k\left[x_{1}, \ldots, x_{n}\right]$, we say that $f$ reduces to zero modulo $G$, written
$$f \rightarrow G 0$$
iff can be written in the form
$$f=a_{1} g_{1}+\cdots+a_{t} g_{t},$$
such that whenever $a_{i} g_{i} \neq 0$, we have
$$\text { multideg }(f) \geq \text { multideg }\left(a_{i} g_{i}\right) \text {. }$$

## 线性代数作业代写linear algebra代考|The Implicitization Problem

$$x=t+u, \quad y=t^{2}+2 t u, z=t^{3}+3 t^{2} u$$

$$-(4 / 3) x^{3} z+x^{2} y^{2}+2 x y z-(4 / 3) y^{3}-(1 / 3) z^{2}=0 .$$

## 线性代数作业代写linear algebra代考|Improvements on Buchberger’s Algorithm

$$f \rightarrow G 0$$
iff 可以写成形式
$$f=a_{1} g_{1}+\cdots+a_{t} g_{t},$$

$$\text { multideg }(f) \geq \text { multideg }\left(a_{i} g_{i}\right)$$

# 计量经济学代写

## 在这种情况下，如何学好线性代数？如何保证线性代数能获得高分呢？

1.1 mark on book

【重点的误解】划重点不是书上粗体，更不是每个定义，线代概念这么多，很多朋友强迫症似的把每个定义整整齐齐用荧光笔标出来，然后整本书都是重点，那期末怎么复习呀。我认为需要标出的重点为

A. 不懂，或是生涩，或是不熟悉的部分。这点很重要，有的定义浅显，但证明方法很奇怪。我会将晦涩的定义，证明方法标出。在看书时，所有例题将答案遮住，自己做，卡住了就说明不熟悉这个例题的方法，也标出。

B. 老师课上总结或强调的部分。这个没啥好讲的，跟着老师走就对了

C. 你自己做题过程中，发现模糊的知识点

1.2 take note

1.3 understand the relation between definitions