  Tate introduced in [Ta71] the notion of Tate algebras to serve, in the
context of analytic geometry over the-adics, as a counterpart of polynomial
algebras in classical algebraic geometry. In [CVV19, CVV20] the formalism of
Gr{\"o}bner bases over Tate algebras has been introduced and advanced
signature-based algorithms have been proposed. In the present article, we
extend the FGLM algorithm of [FGLM93] to Tate algebras. Beyond allowing for
fast change of ordering, this strategy has two other important benefits. First,
it provides an efficient algorithm for changing the radii of convergence which,
in particular, makes effective the bridge between the polynomial setting and
the Tate setting and may help in speeding up the computation of Gr{\"o}bner
basis over Tate algebras. Second, it gives the foundations for designing a fast
algorithm for interreduction, which could serve as basic primitive in our
previous algorithms and accelerate them significantly.
