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| 研究生： |
陳亮 Chen, Liang |
|---|---|
| 論文名稱： |
熱帶亞純函數及其在差分方程之應用 Tropical Meromorphic Functions and Their Application on Difference Equations |
| 指導教授： |
蔡炎龍
Tsai, Yen Lung |
| 學位類別： |
碩士
Master |
| 系所名稱： |
理學院 - 應用數學系 Department of Mathematical Sciences |
| 論文出版年： | 2011 |
| 畢業學年度： | 99 |
| 語文別： | 英文 |
| 論文頁數： | 46 |
| 中文關鍵詞： | 熱帶幾何 、熱帶多項式 、熱帶亞純函數 、熱帶週期函數 |
| 外文關鍵詞： | tropical geometry, tropical polynomial, tropical meromorphic function, tropical periodic function |
| 相關次數： | 點閱：175 下載：58 |
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在這篇論文中，一個熱帶亞純函數(tropical meromorphic function)若給定有限個零根(roots)與極點(poles)還有它們的重數(multiplicities)，我們證明了這個熱帶亞純函數的存在與唯一性。另外，一個熱帶週期函數(tropical periodic function)若給定一個週期區間內的有限個零根與極點還有它們的重數，我們也找到了這個熱帶週期函數的一個簡單表達式。接著，給定一個一階線性差分方程(first-order linear difference equation)，我們討論了各種情況下的所有熱帶亞純函數解的表達式。最後，對於連續函數我們提供了一個它的熱帶近似函數，希望對於解差分方程的熱帶亞純函數解時能有所助益。
In this thesis, we find the formula of tropical meromorphic function by giving finite number of roots and poles (with multiplicities). We also find a simple formula for tropical periodic function by giving finite number of roots and poles (with multiplicities) during a period [0,T). We then discuss all cases of the tropical meromorphic solution functions of first-order linear difference equation. At last, we provide a tropical approximated function of a given continuous function. We hope it is helpful in solving the tropical meromorphic solution functions of a given difference equation.
1 Introduction 2
2 Background 4
2.1 Arithmetic . . . . . . . . . . . . . . . . . . . . 4
2.2 Some Equalities and Inequalities . . . . . . . . . 5
3 Tropical Meromorphic Functions 7
3.1 Tropical Polynomials in One Variable . . . . . . . 8
3.2 Tropical Meromorphic Functions . . . . . . . . . . 10
3.3 Extended Tropical Meromorphic Functions . . . . . 18
4 Tropical Periodic Functions 22
4.1 Generating Functions of Tropical Periodic Functions 23
4.2 R-linear Combination of Tropical 1-Periodic Functions 25
4.3 R-linear Combination of Tropical T-Periodic Functions 29
5 Application on Di erence Equations 35
5.1 Tropical Counterpart to The Exponential Function 36
5.2 Application on Di erence Equations: First Order 38
5.3 Tropical Approximated Function . . . . . . . . . . 42
Bibliography
[1] Nathan Grigg. Factorization of tropical polynomials in one and several variables. Honor's thesis,Brigham Young University, 2007.
[2] Nathan Grigg and Nathan Manwaring. An elementary proof of the fundamental theorem of tropical algebra. arXiv:math.CO/0707.2591, 2008.
[3] Rodney G. Halburd and Neil J. Southall. Tropical nevanlinna theory and ultradiscrete equations. Int. Math. Res. Not. IMRN, (5):887{911, 2009.
[4] Ilpo Laine and Kazuya Tohge. Tropical nevanlinna theory and second main theorem. ArXiv e-prints, October 2009.
[5] Diane Maclagan and Bernd Sturmfels. Introduction to tropical geometry. 2009.
[6] Jurgen Richter-Gebert, Bernd Sturmfels, and Thorsten Theobald. First steps in tropical geometry. ArXiv e-prints:math/0306366, 2003.
[7] David Speyer and Bernd Sturmfels. Tropical mathematics. ArXiv e-prints:math/0408099, 2004.
[8] Yen-Lung Tsai. Working with tropical meromorphic functions of one variable. ArXiv e-prints, January 2011.
