基本素養 Basic Literacy

核心能力 Competence

掌握連續與逼進的能力
Ability to handle continuity phenomena and approximations.
處理數字及符號運算的能力
Ability to operate with symbols and digits.
透視形像與空間的能力
Ability to visualize space, shape, and images.
利用計算機處理數學的能力
Ability of using computer to do mathematics.
應用隨機理論的能力
Ability to apply stochastic mathematics.
處理大量數據的能力
Ability to handle large amount of data.
處理離散型數學的能力
Ability to do mathematics relating to discrete models.
獨立研究的能力
Ability to do independent studies of mathematics.
使用計算機的能力
Abilities to use computer as a tool.

課程概述 Course Description

格羅布納基,布其柏格算法。
Groebner basis, Buchberger's algorithm.

課程學習目標 Course Objectives

  • Grobner bases and Buchberger's algorithm
  • State polytopes and Grobner fans
  • Fundamental theorem of tropical geometry
  • 課程進度 Course Outline

    週次 Week進度說明 Progress Description
    1Polynomial rings and Ideals [EH: Chap 1]
    2Monomial orders and Initial ideals [EH: 2.1-2.2]
    3Division algorithm and Buchberger's criterion [EH: 2.3-2.4]
    4Buchberger's algorithm and Reduced Grobner bases [EH:2.5-2.6]
    5Elimination of variables [EH: 3.1]
    6Applications to operations on ideals [EH:3.2]
    7Zero dimensional ideals and ideals of initial forms [EH: 3.3-3.4]
    8Hilbert's Nullstellensatz and Ideal-variety correspondence [CLO: 4.1~4.4]
    9Primary Decomposition [CLO: 4.5~4.7]
    10Projective varieties [CLO: 8.2~8.3]
    11The Hilbert function and the dimension of a variety [CLO: 9.1~9.3]
    12Newton polytopes, State polytopes, and Grobner fans [St: Chap 2]
    13Fields with valuations [MS: 2.1], Very affine varieties[MS: 2.2]
    14Some polyhedral geometry [MS: 2.3], Grobner bases over fields with valuations [MS: 2.4]
    15Grobner complexes and Tropical Bases [MS: 2.5-2.6]
    16Fundamental theorem of tropical geometry [MS: 3.1-3.2]
    17Structure theorems and the balancing conditions for tropical varieties [MS: 3.3~3.4]
    18Students' Final Reports
     以上每週進度教師可依上課情況做適度調整。The schedule may be subject to change.

    有關課程其他調查 Other Surveys of Courses

    1.本課程是否規劃業界教師參與教學或演講?
    Is there any industry specialist invited in this course? How many times?
    2.本課程是否規劃含校外實習(並非參訪)?
    Are there any internships involved in the course? How many hours?
    3.本課程是否可歸認為學術倫理課程?
    Is this course recognized as an academic ethics course? In the course how many hours are regarding academic ethics topics?
    4.本課程是否屬進入社區實踐課程?
    Is this course recognized as a Community engagement and Service learning course? Which community will be engaged?