We propose a method for tracing implicit real algebraic curves defined by polynomials with rank-deficient Jacobians. For a given curve $f^{-1}(0)$, it first utilizes a regularization technique to compute at least one witness point per connected component of the curve. We improve this step by establishing a sufficient condition for testing the emptiness of $f^{-1}(0)$. We also analyze the convergence rate and carry out an error analysis for refining the witness points. The witness points are obtained by computing the minimum distance of a random point to a smooth manifold embedding the curve while at the same time penalizing the residual of $f$ at the local minima. To trace the curve starting from these witness points, we prove that if one drags the random point along a trajectory inside a tubular neighborhood of the embedded manifold of the curve, the projection of the trajectory on the manifold is unique and can be computed by numerical continuation. We then show how to choose such a trajectory to approximate the curve by computing eigenvectors of certain matrices. Effectiveness of the method is illustrated by examples.

中文翻译：

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秩不足多项式系统的同伴曲线跟踪方法

我们提出了一种跟踪具有秩不足的雅可比行列式的多项式定义的隐式实数代数曲线的方法。对于给定的曲线$ f ^ {-1}（0）$，它首先利用正则化技术为曲线的每个连接分量计算至少一个见证点。我们通过建立足够的条件来测试$ f ^ {-1}（0）$的空度来改进此步骤。我们还分析了收敛速度，并进行了误差分析以完善见证点。通过计算随机点到嵌入曲线的平滑流形的最小距离来获得见证点，同时惩罚局部最小值处的$ f $的残差。为了从这些见证点开始追踪曲线，我们证明了如果将随机点沿着曲线的嵌入式歧管的管状邻域内的轨迹拖动，轨迹在歧管上的投影是唯一的，并且可以通过数值连续来计算。然后，我们展示如何通过计算某些矩阵的特征向量来选择这样的轨迹来近似曲线。实例说明了该方法的有效性。