关于热带格罗布纳基的FGLM算法
On FGLM Algorithms with Tropical Gröbner bases
摘要：
设K是一个具有赋值的域。K上的热带变种可以用考虑K估值的Gr{ö}bner基理论来定义。由于赋值的使用，热带Gr{<$}bner基理论已被证明为p进域上的多项式环上的计算提供了比经典Gr{<$}bner基更稳定的设置。在这篇文章中，我们研究如何FGLM的变化排序算法可以适应热带设置。由于考虑了多项式系数的赋值，经典的FGLM算法的逐单式递增方式计算乘法矩阵和基矩阵的变换根本不能转置到热带环境。我们通过开发新的线性代数算法来缓解这个问题，并将其应用于我们新的热带FGLM算法。动机是双重的。首先，为了计算热带变种，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{<$}bner基。对于维数为0的理想，热带FGLM算法提供了一种有效的方法，从一个权重的热带Gr{<$}bner基到另一个权重的热带Gr{<$}bner基。其次，FGLM策略可以应用于从热带Gr{<$}bner基到经典Gr{<$}bner基的转换。我们提供了工具来链接热带Gr{<$}bner基的稳定计算（对于权重[0，. ..，0]）与[RV 16]的FGLM的p-adic稳定化变体一起计算字典或形状位置基础。我们所有的算法都已经在SageMath中实现。我们提供数值例子来说明时间复杂度。然后，我们说明了我们的策略的优越性，关于稳定的p-adic数值计算。
英文摘要：
Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Gr{ö}bner bases taking into account the valuation of K. Because of the use of the valuation, the theory of tropical Gr{ö}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{ö}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{ö}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{ö}bner basis from one weight to one for another weight. Secondly, the FGLM strategy can be applied to go from a tropical Gr{ö}bner basis to a classical Gr{ö}bner basis. We provide tools to chain the stable computation of a tropical Gr{ö}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.
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