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 \citelib{MLib} that implements this algorithm. Using this library and about 723 000 core-hours on the HPC center \textit{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，用于计算与曼德勃罗集 $\mathcal{M}$ 相关的两类多项式的所有根：这些多项式的根分别对应于 $f_c(z)=z^2+c$ 中周期临界动力学（双曲中心）的参数 $c\in\mathcal{M}$ ，或前周期动力学（Misiurewicz-Thurston参数）的参数。该算法基于离散等高线的计算，能为牛顿法提供极优的初始值。实践中观察到，这些多项式可在度数的线性时间内完成分割。本文附有实现该算法的代码库\citelib{MLib}。利用该库及HPC中心\textit{Rom\'eo}（兰斯）约723 000核时，我们成功找到了周期 $\leq 41$ 的所有双曲中心，以及周期与前周期之和 $\leq 35$ 的所有Misiurewicz-Thurston参数。具体而言，该任务涉及分割一个万亿次多项式，即度数 $\sim10^{12}$ 的多项式，比先前最优技术领先数个数量级。此外还需处理数值结果的可认证性问题——我们对此从数学和生产线两个层面进行了详细探讨。认证数据库已向科学界开放。对于仅需硬件算术（FP80浮点数）即可表示的较小周期，我们的算法实现可在不到一个核日内分割对应的 $\sim10^{9}$ 次多项式。我们还通过根间距的统计分析补充了这些基准测试，证实该类族中其他多项式的分割必须使用更高精度算术。