标题： 无限欧几里得距离判别式 显示英文标题

标题： Infinite Euclidean Distance Discriminants

摘要： 我们研究代数簇的无限欧几里得距离判别式，其定义为在从欧几里得距离对应关系的第二个投影下，纤维为高维的数据点的轨迹。 特别是，这些判别式包含所有对于最近点问题具有无限多个临界点的数据点。 我们提供了计算无限欧几里得距离判别式的计算机代码，并利用它展示了大量具有非空此类判别式的代数簇。 此外，我们证明了对于任何数据点，第二个投影下的纤维被包含在一个以该点为中心的超球面的有限并集中。 对于曲线，我们给出了完整的刻画；它们的无限欧几里得距离判别式被证明是仿射线性空间。 最后，我们在三维空间中引入并刻画了斜管曲面。 根据构造，这些曲面具有一个一维的无限欧几里得距离判别式。 我们进一步证明了许多斜管曲面的欧几里得距离次数明显低于同次数的一般曲面。 显示英文摘要

摘要： We study infinite Euclidean distance discriminants of algebraic varieties, defined as the loci of data points whose fibers under the second projection from the Euclidean distance correspondence are positive-dimensional. In particular, these discriminants contain all data points with infinitely many critical points for the nearest-point problem. We present computer code that computes the infinite Euclidean distance discriminant, and use it to present numerous varieties with nonempty such discriminants. Moreover, we prove that for any data point, the fiber under the second projection is contained in a finite union of hyperspheres centered at that point. For curves, we include a complete characterization; their infinite Euclidean distance discriminants turn out to be affine linear spaces. Finally, we introduce and characterize skew-tube surfaces in three-dimensional space. By construction, these have a one-dimensional infinite Euclidean distance discriminant. We further demonstrate that many skew-tubes have significantly lower Euclidean distance degrees than generic surfaces of the same degree.