  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.
