熱帶曲線之圖形化研究

Abstract

熱帶曲線(tropical curves) 是定義在熱帶半環(tropical semiring) 上的代數曲線。熱帶曲線是古典代數曲線經由某些賦值(valuation) 的映像，所以許多重要的代數曲線性質也同樣發生在熱帶曲線上。本篇論文我們試著將熱帶曲線圖形化。\r\n首先，我們根據熱帶曲線的理論發展出幾個繪出熱帶曲線的演算法。再者，我們以電腦程式語言Python 去實現這些算演算法。我們發展的是跨平台的程式碼，可以在Linux, Mac OS X, Windows 等作業系統執行。

Tropical curves are algebraic curves dened over the tropical semiring.\r\nThey are the images of classical algebraic curves under some valuation maps, so reect many important properties of classical algebraic curves. In this thesis,\r\nwe try to visualize tropical curves. We study the theory of tropical curves and develop several algorithms to draw the graphs of tropical curves. \r\nFurthermore, we implement these algorithms in Python programming language. These codes are cross-platform, running on Linux, Mac OS X, and Windows.

1 Introduction... 1\r\n2 Plane tropical curves... 3\r\n2.1 Tropical curves as limit of amoebas... 3\r\n2.2 Tropical curves via varieties over the eld of Puiseux series... 6\r\n2.3 Tropical curves as varities over the max-plus semiring... 12\r\n3 Arithmetic of the max-plus semiring... 15\r\n4 Tropical polynomial... 18\r\n5 Curves... 26 \r\n5.1 Hypersurface... 26\r\n5.2 Tropical curves as balanced graphs... 28\r\n6 Draw tropical curves by Python... 30