熱帶導數與熱帶反導數

Abstract

在這篇論文中,我們定義了熱帶導數和熱帶反導數.當我們對兩個相同的熱帶多項式求導數時,可能會得到不同的函數.為了克服此困難,我們限制在最大係數多項式下才求導數.熱帶導數的定義與古典導數相當不同.特別的是,我們有d/dxan⊙x^(⊙n)= an⊙x⊙n-1.將它線性化,我們得到d/dx[an⊙x^(⊙n)⊕an-1⊙x⊙n-1 ⊕…. a1⊙x⊕a0] = an⊙x⊙n-1 ⊕an-1⊙x⊙n-2⊕…⊕a1.我們將會解釋為什麼使用這種定義.導數對了解熱帶幾何很有幫助,它也引出了一些與古典導數相似的資訊.最後,我們討論如何定義及求熱帶多項式的熱帶反導數

In this thesis, we define the tropical derivatives and anti-derivatives. When we differ-\nentiate two identical tropical polynomials, we might get two different functions. In order to overcome the diffculties, we restrict the polynomials to largest coeffcient polynomials to avoid unpredictable results when taking derivatives. The definitiion of the tropical derivatives is quite diffrent from the definition of classical derivatives. In particular, we have d/dxan⊙x^(⊙n)= an⊙x⊙n-1 . To extend it linearly, we obtain d/dx[an⊙x^(⊙n)⊕\na n-1⊙x⊙n-1 ⊕…. a1⊙x⊕a0] = an⊙x⊙n-1 ⊕a n-1⊙x⊙n-2⊕…⊕a1. We will explain why we use this kind of definition. The derivatives are helpful in understanding more about tropical geometry, and it carries out some information similar to classical derivatives. Finally, we discuss how to define and find tropical anti-derivatives for tropical polynomials.\nKeywords : Tropical derivatives, tropical anti-derivatives, tropical polynomials.