  We introduce a method to determine the maximum nilpotent orbit which
intersects a variety of nilpotent matrices described by a strictly upper
triangular matrix over a polynomial ring. We show that the result only depends
on the ranks of its submatrices and we introduce conditions on a subvariety so
that it intersects the same orbit. Then we describe a maximal nilpotent
subalgebra of the centralizer of any nilpotent matrix; the previous method
allows us to show that the maximum nilpotent orbit which intersects that
centralizer only depends on which entries are identically 0 in that subalgebra.
The aim of the paper is to prove a simple algorithm for the determination of
the maximum nilpotent orbit which intersects that centralizer, which was
conjectured by Polona Oblak.
