The equivalence problem for deterministic pushdown transducers with inputs in a free monoid
and outputs in a linear group 
, is shown to be decidable.
We first reduce the problem to the equivalence problem for deterministic rational series on the alphabet associated with a dpdt
and with coefficients in the group H. We then exhibit a formal
system which generates only equivalent pairs of such series. Conversely,
every pair of equivalent series admits a regular proof. Using previous
works on test-sets, we show that the set of ``regular'' proofs is recursively
enumerable.
As a corollary, the equivalence problem for dpdt's from a free monoid
into a free group 
(or, a fortiori, a free monoid 
), is decidable.