一个新的计算平面有理曲线$\mu$基的算法

A New Algorithm for Computing $\mu$-Bases of Planar Rational Curves

摘要: $\mu$ 基作为研究平面有理曲线的重要工具，不仅能够刻画参数曲线的动直线结构，还可用于曲线隐式化、奇异点分析以及参数形式与隐式形式之间的相互转化。现有$\mu$ 基的计算方法大多以多项式向量的合冲模为基本对象，并通过高斯消元法、矩阵行化简或多项式矩阵分解等方法实现计算，具有较强的一般性和较好的算法效率。本文从另一角度出发，利用重新定义的向量多项式，提出一种构造$\mu$ 基的算法，并辅以实例说明。

Abstract: As an important algebraic tool for studying planar rational curves, a $\mu$ -basis not only captures the moving-line structure of a parametric curve, but also plays an essential role in deriving implicit equations, analyzing singularities, and converting between parametric and implicit forms. Existing methods for computing $\mu$ -bases usually take the syzygy module of a polynomial vector as the underlying object, and compute them using Gaussian elimination, matrix row reduction, or polynomial matrix factorization. These methods have broad applicability and high computational efficiency. In this paper, we approach the problem from a different perspective. By introducing a new class of vector polynomials, we propose an algorithm for constructing $\mu$ -bases and illustrate the method with examples.