  We study a relationship between rational proper maps of balls in different
dimensions and strongly plurisubharmonic exhaustion functions of the unit ball
induced by such maps. Putting the unique critical point of this exhaustion
function at the origin leads to a normal form for rational proper maps of
balls. The normal form of the map, which is up to composition with unitaries,
takes the origin to the origin, and it normalizes the denominator by
eliminating the linear terms and diagonalizing the quadratic part. The singular
values of the quadratic part of the denominator are spherical invariants of the
map. When these singular values are positive and distinct, the normal form is
determined up to a finite subgroup of the unitary group. We also study which
denominators arise for cubic maps, and when we do not require taking the origin
to the origin, which maps are equivalent to polynomials.

Complex Variables