  In characteristic zero, the residual order constitutes, after the local
multiplicity, the second key invariant for the resolution of singularities. It
is defined as the order of the coefficient ideal in a local hypersurface of
maximal contact, minus the exceptional multiplicities. It does not increase
under blowup in permissible centers as long as the local multiplicity remains
constant. In positive characteristic, however, the residual order (defined now
as the maximum over all smooth local hypersurfaces) may increase under blowup.
In the article we analyze in detail the circumstances when this happens. This
may help to develop a modification of the residual order which does work in
positive characteristic.
