  For two ideals $I$ and $J$ of a noetherian ring, we characterize, in terms of
the vanishing of Tor modules, when the associated graded ring of the sum $I+J$
is isomorphic to the tensor product of the associated graded ring of $I$ and
the associated graded ring of $J$. It is shown that the relation type of the
tensor product of two standard algebras is bounded above by the maximum of the
relation type of each algebra. As a consequence, we deduce a uniform bound for
the relation type of maximal ideals of an excellent ring and a classical result
of Duncan and O'Carroll on the strong uniform Artin-Rees property.
