Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Gr{\"o}bner bases taking into account the valuation of K. Because of the use of the valuation, the theory of tropical Gr{\"o}bner bases has proved to provide settings for computations over polynomial rings over a p-adic field that are more stable than that of classical Gr{\"o}bner bases. In this article, we investigate how the FGLM change of ordering algorithm can be adapted to the tropical setting. As the valuations of the polynomial coefficients are taken into account, the classical FGLM algorithm's incremental way, monomo-mial by monomial, to compute the multiplication matrices and the change of basis matrix can not be transposed at all to the tropical setting. We mitigate this issue by developing new linear algebra algorithms and apply them to our new tropical FGLM algorithms. Motivations are twofold. Firstly, to compute tropical varieties, one usually goes through the computation of many tropical Gr{\"o}bner bases defined for varying weights (and then varying term orders). For an ideal of dimension 0, the tropical FGLM algorithm provides an efficient way to go from a tropical Gr{\"o}bner basis from one weight to one for another weight. Secondly, the FGLM strategy can be applied to go from a tropical Gr{\"o}bner basis to a classical Gr{\"o}bner basis. We provide tools to chain the stable computation of a tropical Gr{\"o}bner basis (for weight [0,. .. , 0]) with the p-adic stabilized variants of FGLM of [RV16] to compute a lexicographical or shape position basis. All our algorithms have been implemented into SageMath. We provide numerical examples to illustrate time-complexity. We then illustrate the superiority of our strategy regarding to the stability of p-adic numerical computations.

中文翻译：

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基于热带 Gr\"obner 基的 FGLM 算法

设 K 是一个带有估值的字段。考虑到 K 的估值，可以用 Gr{\"o}bner 基理论来定义 K 上的热带品种。由于使用了估值，热带 Gr{\"o}bner 基理论已经证明为 p-adic 域上多项式环的计算提供设置，该设置比经典 Gr{\"o}bner 基更稳定。在本文中，我们研究了 FGLM 排序算法的变化如何适应热带设置. 由于考虑了多项式系数的估值，经典FGLM 算法的增量方式，单项逐项，计算乘法矩阵和基矩阵的变化根本不能转置到热带环境中。我们通过开发新的线性代数算法来缓解这个问题，并将它们应用到我们新的热带 FGLM 算法中。动机是双重的。首先，为了计算热带变体，人们通常要计算许多为不同权重（然后是不同项阶）定义的热带 Gr{\"o}bner 基。对于维度 0 的理想，热带 FGLM 算法提供了一种有效的从一个重量到另一个重量的热带 Gr{\"o}bner 基础的方法。其次，FGLM 策略可以应用于从热带 Gr{\"o}bner 基础到经典 Gr{\"o}bner 基础。我们提供了将热带 Gr{\"o}bner 基础（对于权重 [0,.., 0]）的稳定计算与 [RV16] 的 FGLM 的 p-adic 稳定变体链接起来的工具，以计算字典序或形状位置基础。我们所有的算法都已在 SageMath 中实现。我们提供了数值例子来说明时间复杂性。然后，我们说明了我们的策略在 p-adic 数值计算稳定性方面的优越性。