论文标题
代数曲线的签名通过数值代数几何形状
Signatures of algebraic curves via numerical algebraic geometry
论文作者
论文摘要
我们将数值代数几何形状应用于检测两个平面代数曲线之间对称性的不变理论问题。我们描述了一个有效的平等测试，该测试用“概率 - 一个”确定，两个有理图是否具有相同的图像，直到Zariski闭合。不变理论的应用是基于与在各个曲线上线性作用的组相关的合适的签名图的构建。我们考虑了这种结构的两个版本：差分和关节标志地图。在我们的示例和计算实验中，我们专注于复杂的欧几里得群体，并引入了代数关节签名，我们证明，在此动作和曲线对称组的大小下，我们证明了曲线的等价。我们证明该测试是有效的，并将其用于经验将差分和关节特征与不同类型的噪声的灵敏度进行比较。
We apply numerical algebraic geometry to the invariant-theoretic problem of detecting symmetries between two plane algebraic curves. We describe an efficient equality test which determines, with "probability-one", whether or not two rational maps have the same image up to Zariski closure. The application to invariant theory is based on the construction of suitable signature maps associated to a group acting linearly on the respective curves. We consider two versions of this construction: differential and joint signature maps. In our examples and computational experiments, we focus on the complex Euclidean group, and introduce an algebraic joint signature that we prove determines equivalences of curves under this action and the size of a curve's symmetry group. We demonstrate that the test is efficient and use it to empirically compare the sensitivity of differential and joint signatures to different types of noise.
