本論文主要是研究凱萊-巴拉赫定理的不同證明及其推廣，從而了解經典代數幾何的一些主題，如獨立條件、馬克斯·諾特基本定理和除子。我們還介紹了一些有趣又相關的題材。

The main propose is studying different proofs of Cayley-Bacharach theorem and its extension, thereby understanding some topics of classical algebraic geometry, such as independent conditions, Max Noether's fundamental theorem, and divisors. We also present some interesting related topics.

1. Introduction---------------------------------------------5

2. Background-----------------------------------------------6
2.1 Algebraic Curves----------------------------------------6
2.2 Mapping between Affine and Projective planes------------8
2.3 Intersection of Curves----------------------------------9
2.4 Pappus’s Theorem---------------------------------------10

3. Algebraic structure of Curves---------------------------12
3.1 Linear System of Homogeneous Polynomials---------------12
3.2 Conditions Determined by Points------------------------14
3.3 Cayley–Bacharach Theorem-------------------------------18

4. Fundamental Theorem of Algebraic Functions--------------21
4.1 Conditions Determined by Intersection Points-----------21
4.2 Application of AF+BG Theorem---------------------------23
4.3 Related Theorems---------------------------------------25

5. Algebraic structure of Points---------------------------27
5.1 Divisors-----------------------------------------------27
5.2 Vector Space of Rational Functions---------------------30
5.3 CB4 - CB5----------------------------------------------32

6. Further Study-------------------------------------------36

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