  We consider polynomial maps, which we call degree $d$-linear maps, that
satisfy the Jacobian condition. We prove that certain infinite families of
elements, which appear in the coefficients of the formal inverse of such maps,
are in the ideal determined by the Jacobian condition. Using the
Cayley-Hamilton theorem, we provide expressions for these elements in terms of
the generators of that ideal. We also give a combinatorial proof of the
Cayley-Hamilton theorem similar to that of Straubing \cite{Straubing}. We also
include results of Gr\"obner basis computations regarding other elements.
