Topological data analysis (TDA) provides a growing body of tools for
computing geometric and topological information about spaces from a finite
sample of points. We present a new adaptive algorithm for finding provably
dense samples of points on real algebraic varieties given a set of defining
polynomials. The algorithm utilizes methods from numerical algebraic geometry
to give formal guarantees about the density of the sampling and it also employs
geometric heuristics to reduce the size of the sample. As TDA methods consume
significant computational resources that scale poorly in the number of sample
points, our sampling minimization makes applying TDA methods more feasible. We
provide a software package that implements the algorithm and also demonstrate
the implementation with several examples.

本文介绍了一种新的自适应算法，可以在给定定义多项式集的情况下，在实数代数多项式上找到证明的点样本。该算法利用了数值代数几何的方法，可以形式化保证采样的密度，并利用几何启发式方法减少采样的大小。结果表明，该算法可以使得使用 TDA 方法更加可行。

用于拓扑数据分析的实代数变量采样

Sampling real algebraic varieties for topological data analysis

We determine the algebraic degree of minimal problems for the calibrated
trifocal variety in computer vision. We rely on numerical algebraic geometry
and the homotopy continuation software Bertini.

利用数值代数几何和拓扑学软件 Bertini，确定了在计算机视觉中用于校准三焦点变量的最小问题的代数度。

标定三基点解析式的最小问题

Minimal Problems for the Calibrated Trifocal Variety

Techniques from representation theory, symbolic computational algebra, and
numerical algebraic geometry are used to find the minimal generators of the
ideal of the trifocal variety. An effective test for determining whether a
given tensor is a trifocal tensor is also given.

运用表示论、符号计算代数和数值代数几何技术，找出三视点变量理想的最小生成元，并提供确定给定张量是否为三视点张量的有效测试。