This article presents a new algorithm to compute all the roots of two families of polynomials that are of interest for the Mandelbrot set $\mathcal{M}$ : the roots of those polynomials are respectively the parameters $c\in\mathcal{M}$ associated with periodic critical dynamics for $f_c(z)=z^2+c$ (hyperbolic centers) or with pre-periodic dynamics (Misiurewicz-Thurston parameters). The algorithm is based on the computation of discrete level lines that provide excellent starting points for the Newton method. In practice, we observe that these polynomials can be split in linear time of the degree. This article is paired with a code library [Mandel] that implements this algorithm. Using this library and about 723 000 core-hours on the HPC center Rom\'eo (Reims), we have successfully found all hyperbolic centers of period $\leq 41$ and all Misiurewicz-Thurston parameters whose period and pre-period sum to $\leq 35$. Concretely, this task involves splitting a tera-polynomial, i.e. a polynomial of degree $\sim10^{12}$, which is orders of magnitude ahead of the previous state of the art. It also involves dealing with the certifiability of our numerical results, which is an issue that we address in detail, both mathematically and along the production chain. The certified database is available to the scientific community. For the smaller periods that can be represented using only hardware arithmetic (floating points FP80), the implementation of our algorithm can split the corresponding polynomials of degree $\sim10^{9}$ in less than one day-core. We complement these benchmarks with a statistical analysis of the separation of the roots, which confirms that no other polynomial in these families can be split without using higher precision arithmetic.

翻译：本文提出一种新算法，用于计算与曼德博集合$\mathcal{M}$相关的两类多项式族的所有根：这些多项式的根分别对应$f_c(z)=z^2+c$中具有周期临界动力学（双曲中心点）或预周期动力学（米修列维奇-瑟斯顿参数）的参数$c\in\mathcal{M}$。该算法基于离散等值线的计算，为牛顿法提供了优异的初始点。实际应用中，我们观察到这些多项式可在与次数成线性关系的时间内完成分解。本文配套提供实现该算法的代码库[Mandel]。借助该代码库并利用高性能计算中心Roméo（兰斯）约723,000核时的计算资源，我们成功找到了所有周期$\leq 41$的双曲中心点，以及周期与预周期之和$\leq 35$的所有米修列维奇-瑟斯顿参数。具体而言，该任务涉及分解次数达$\sim10^{12}$的万亿次多项式，这比现有技术水平高出数个数量级。同时需要解决数值结果的可靠性认证问题，我们从数学角度和生产流程层面对此进行了详细探讨。已认证的数据库可供科学界使用。对于仅需硬件算术（FP80浮点运算）即可表示的较小周期，本算法实现可在单核单日内完成对应次数$\sim10^{9}$多项式的分解。我们通过根间距的统计分析补充了这些基准测试，结果证实：若不采用更高精度算术，这些多项式族中不存在其他可分解的多项式。