  We give an upper bound for the maximal slope of the tensor product of several
non-zero Hermitian vector bundles on the spectrum of an algebraic integer ring.
By Minkowski's theorem, we need to estimate the Arakelov degree of an arbitrary
Hermitian line subbundle $\bar M$ of the tensor product. In the case where the
generic fiber of $M$ is semistable in the sense of geometric invariant theory,
the estimation is established by constructing, through the classical invariant
theory, a special polynomial which does not vanish on the generic fibre of $M$.
Otherwise we use an explicte version of a result of Ramanan and Ramanathan to
reduce the general case to the former one.
