发表机构
Institute of Science and Technology Austria (ISTA)（奥地利科学技术研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究如何计算多项式系统奇异解，提出“弧长终局法”用于秩亏1系统，有超线性收敛且计算简便；对秩亏更大系统提出启发式方法，实验效果好；还给出估计曲线普儒斯级数的新稳定方法及\(i\geq2\)时的估计方法。
AI 中文摘要
在数值代数几何中，多项式系统的孤立解通常通过追踪同伦方程定义的解曲线来计算。在接近奇异根时，追踪问题变得极具挑战性。现有方法包括幂级数终局法、柯西终局法和各种基于对偶空间收缩来正则化系统的方法。贡献如下：（1）对于秩亏1系统，引入新的“弧长终局法”，结合伪弧长延拓法思想与曲线普儒斯级数估计，证明在根的邻域超线性收敛，且仅用系统及其雅可比矩阵求值。（2）对于秩亏更大的系统，提出启发式“提升弧长终局法”，实验结果良好。（3）给出估计曲线普儒斯级数的新方法，比之前更稳定，还能估计\(i\geq2\)时的\(k_i/c\)。
英文摘要
In Numerical Algebraic Geometry (NAG) isolated solutions of polynomial systems are usually computed by tracking a solution curve defined by a homotopy equation. The tracking problem becomes especially challenging close to a singular root (the ``endgame'' regime). Existing approaches include power series endgames, Cauchy endgames, and various methods that regularize the system via dual-space-based {\em deflation}. We introduce a new ``Arclength Endgame'' for corank-1 systems which combines the idea of the classical {\em pseudo-arclength continuation method} with the estimation of the Puiseux series of the curve. We formally prove that it has an R-superlinear rate of convergence in some neighborhood of the root. The method uses only evaluations of the system and its Jacobian. In contrast, previous homotopy techniques with proven superlinear convergence (such as deflation) require computing additional derivatives of the system, or need additional nondegeneracy assumptions involving higher-order derivatives of the system at the singular root. A key step in our approach (as well as in the standard power series endgame) is estimating the ratio $k_1/c$ of the Puiseux series of the curve. We present a new method for that, and discuss its advantages over previous methods.
