臺灣博碩士論文加值系統

熱帶幾何(tropical geometry)在近年來引起數學家的注意，因為它可以簡化許多數學難題。本篇論文主要在探討單變數熱帶多項式(single variable tropical polynomial)的因式分解。對於每個熱帶多項式，我們都可以定義其對應的最大係數熱帶多項式(largest-coefficient tropical polynomial )，而且此最大係數熱帶多項式可以因式分解為線性乘積。根據此結果，熱帶代數基本定理(Fundamental Theorem of Tropical Algebra)即成立。此外，可將單變數熱帶多項式因式分解的許多概念延伸至多變數的情形。

Tropical geometry draw much attention recent years for it simplifies many difficult classical mathematics problems. The thesis mainly discuss factorization of single variable tropical polynomials. For every tropical polynomial, we define the corresponding largest-coefficient tropical polynomial. We show that each largest-coefficient tropical polynomial can be factorized into a product of linear terms. As a result, the Fundamental Theorem of Tropical Algebra holds. Furthermore, we observe that many notions of factorization of single variable tropical polynomial can be extended to multivariate cases.

1 Introduction .....................................2
2 Background .....................................5
2.1 Arithmetic .....................................5
2.2 Tropical Polynomial in one variable ............6
3 Fundamental Theorem of Tropical Algebra ..........9
3.1 Tropical Polynomials and Tropical Functions ...10
3.2 The Fundamental Theorem of Tropical Algebra ...21
3.3 The Corner Locus of a Polynomial ..............23
4 Conclusion ......................................26

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