  We prove a sharp degree bound for polynomials constant on a hyperplane with a
fixed number of nonnegative distinct monomials. This bound was conjectured by
John P. D'Angelo, proved in two dimensions by D'Angelo, Kos and Riehl and in
three dimensions by the authors. The current work builds upon these results to
settle the conjecture in all dimensions. We also give a complete description of
all polynomials in dimensions 4 and higher for which the sharp bound is
obtained. The results prove the sharp degree bounds for monomial CR mappings of
spheres in all dimensions.
