Power-exact, nilpotent, homogeneous matrices
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We obtain the formula for the inverse of polynomial maps of of the form where M(x) is a homogeneous of degree m nilpotent matrix, all of whose powers are exact; and we obtain recursion relations for the scalars cmk . Such maps have recently been considered by Connell and Zweibel, but our derivations is based on our earlier result that F −1(a) where x(t z a) is the unique solution, with initial condition x(0z a)=z, of the Ważewski differential equation dx/dt=Ft (x)−1 a=a−M(x)a, with vector parameter a. Basic to our method is the multilinear matrix function B(x y,…z) uniquely determined by M(x). We give a new proof that all powers of M are exact provided only that both M and M 2 are exact.