  The constraint nondegeneracy condition is one of the most relevant and useful
constraint qualifications in nonlinear semidefinite programming. It can be
characterized in terms of any fixed orthonormal basis of the, let us say,
$\ell$-dimensional kernel of the constraint matrix, by the linear independence
of a set of $\ell(\ell+1)/2$ derivative vectors. We show that this linear
independence requirement can be equivalently formulated in a smaller set, of
$\ell$ derivative vectors, by considering all orthonormal bases of the kernel
instead. This allows us to identify that not all bases are relevant for a
constraint qualification to be defined, giving rise to a strictly weaker
variant of nondegeneracy related to the global convergence of an external
penalty method. We use some of these ideas to revisit an approach of Forsgren
[Math. Prog. 88, 105--128, 2000] for exploiting the sparsity structure of a
transformation of the constraints to define a constraint qualification, which
led us to develop another relaxed notion of nondegeneracy using a simpler
transformation. If the zeros of the derivatives of the constraint function at a
given point are considered, instead of the zeros of the function itself in a
neighborhood of that point, we obtain an even weaker constraint qualification
that connects Forsgren's condition and ours.
