The monic rank

We introduce the monic rank of a vector relative to an affinehyperplane 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 · e is the sum of d dth 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.