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研究生： 李詩淳
Shih-Chun Li
論文名稱： An application of Bezout's theorem: the effective minimal intersection number of a plane curve
指導教授： 陳正傑
Jheng-Jie Chen
口試委員:
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系
Department of Mathematics
論文出版年： 2020
畢業學年度： 108
語文別： 中文
論文頁數： 38
中文關鍵詞： 仿射平面曲線 、交點數 、Bezout定理 、近似根
外文關鍵詞： Embedding line, Bezout's Theorem, intersection number, approximate roots, algebraic curve
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    • 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)

    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).

    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

    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

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