An application of Bezout′s theorem: the effective minimal intersection number of a plane curve

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李詩淳(Shih-Chun Li) 查詢紙本館藏

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論文名稱

(An application of Bezout′s theorem: the effective minimal intersection number of a plane curve)

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摘要(中)

這篇碩士論文要是研究仿射平面曲線的交點數。事實上，我們將張海潮教授和王立中教授在[CW]的論述中，歸納並得出以下我們的主要定理:
「如果曲線F(1,y,z)在無窮遠處只有一個place，則我們可以建構出與曲線F(1,y,z)相交的曲線G_j，使得它們在所有曲線上達到最小的正交點數。」

這是應用到Bezout定理，以及在[Moh1, Moh2, Moh3, Moh4]介紹的近似根概念。此外，我們可以將Embedding Line Theorem作為一個應用並加以證明。(請參閱第八章)

摘要(英)

In this thesis, we study the intersection number of affine plane curves.
Actually, we generalize the argument of Chang and Wang in [CW] to obtain our main theorem as follows:

“if the curve $F(1,y,z)$ has only one place at infinity, then we would construct a curve G_j which intersects curve F(1,y,z) attaining the positive minimal intersection number among all curves."

This is an application of Bezout′s Theorem and the approximate roots introduced by [Moh1, Moh2, Moh3, Moh4].

Besides, we can reprove the Embedding Line Theorem as an application (see section 8).

關鍵字(中)

★ 仿射平面曲線
★ 交點數
★ Bezout定理
★ 近似根

關鍵字(英)

★ Embedding line
★ Bezout′s Theorem
★ intersection number
★ approximate roots
★ affine
★ algebraic curve

論文目次

1 Introduction........................................1
2 Basic Knowledge of Commutative Algebra..............2
2.1 Ideals and Modules.................................2
2.2 Discrete Valuation Ring............................3
3 Fundamental Knowledge of Algebraic Curves...........4
3.1 Affine Algebraic Sets and Affine Varieties.........4
3.2 The Intersection Properties of Affine Plane Curves.5
3.3 Projective Varieties...............................6
3.4 Bezout’s Theorem...................................8
4 Parametrizations and Places........................14
4.1 Parametrizations of Curves........................14
4.2 Places of Curves..................................15
4.3 Discussion and Example............................15
5 Zariski’s Works....................................18
6 The Approximate Root of Polynomials................20
6.1 Definitions.......................................20
6.2 Applications of Polynomials.......................20
7 Main Theorem.......................................24
8 Embedding Line Theorem.............................27
9 Appendix...........................................29
Reference..............................................30

參考文獻

Newton-Puiseux expansion and generalized Tschirnhausen transformation. I, II;
Embeddings of the line in the plane;
Lectures on expansion techniques in algebraic geometry;
On equisingularity, analytical irreducibility and embedding line theorem;
An Intersection Theoretical Proof of the Embedding Line Theorem;
Algebraic Curves : An introduction to Algebraic Geometry;
Algebraic Geometry;
On Abhyankar-Moh′s epimorphism theorem:
Embeddings of the plane;
Commutative Ring Theory;
Curves on Rational and Irrational Surfaces;
On the concept of approximate roots for algebra;
On characteristic pairs of algebroid plane curves for characteristic p;
On two fundamental theorems for the concept of approximate roots;
Algebra 3rd ed.;
An Algebraic Introduction to Complex Projective Geometry : Commutative Algebra;
Algebraic Curves;
Le problème des modules pour les branches planes;
Commutative Algebra

指導教授

陳正傑(Jheng-Jie Chen)

審核日期

2020-7-28

推文