# 代數導論二 Week 5 (Part 1) - Definition of Polynomial Rings

[TOC]

## 定義：多變數的多項式

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定義：

$$
R[x_1 \dots x_n] = (R[x_1 \dots x_{n-1}])[x_n]
$$

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比如說

$$
R[x_1, x_2] = (R[x_1])[x_2]
$$

### 定義：多變數多項式的次數

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The polynomial ring in the variables $x_1 \dots x_n$ with coefficients in $R$ is denoted by

$$
R[x_1 \dots x_n]
$$

A monic term $(x_1^{d_1} \dots x_n^{d_n})$ is called a mononomial, and $(d_1 + \dots + d_n)$ is called the degree of the term, and the tuple $(d_1, d_2 \dots d_n)$ is called the multidegree of it.

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### 定義：Homogeneous Polynomial

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A polynomial is called homogeneous if all its terms have the same degree. For example

$$
f(x, y) = x^3 + x^2y + y^3
$$
Equivalently, if $f$ satisfies
$$
f(\lambda x_1, \dots, \lambda x_n) = \lambda^d f(x_1,\dots,x_n)
;\quad \forall\lambda \in R$$, then we say f is homogeneous of degree $d$.

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The sum of all its mononial terms of $\deg k$ is called the homogeneous component of $\deg k$ of $f(x_1 \dots x_n)$

$$
\deg f = \text{ largest deg of its monimial terms}
$$

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