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

Exhaustion functions and normal forms for proper maps of balls