  Working over an infinite field of positive characteristic, an upper bound is
given for the nilpotency index of a finitely generated nil algebra of bounded
nil index $n$ in terms of the maximal degree in a minimal homogenous generating
system of the ring of simultaneous conjugation invariants of tuples of $n$ by
$n$ matrices. This is deduced from a result of Zubkov. As a consequence, a
recent degree bound due to Derksen and Makam for the generators of the ring of
matrix invariants yields an upper bound for the nilpotency index of a finitely
generated nil algebra that is polynomial in the number of generators and the
nil index. Furthermore, a characteristic free treatment is given to Kuzmin's
lower bound for the nilpotency index.
