  We introduce the monic rank of a vector relative to an affine-hyperplane
section of an irreducible Zariski-closed affine cone $X$. We show that the
monic rank is finite and greater than or equal to the usual $X$-rank. We
describe an algorithmic technique based on classical invariant theory to
determine, in concrete situations, the maximal monic rank. Using this
technique, we establish three new instances of a conjecture due to B. Shapiro
which states that a binary form of degree $d\cdot e$ is the sum of $d$ $d$-th
powers of forms of degree $e$. Furthermore, in the case where $X$ is the cone
of highest weight vectors in an irreducible representation---this includes the
well-known cases of tensor rank and symmetric rank---we raise the question
whether the maximal rank equals the maximal monic rank. We answer this question
affirmatively in several instances.
