Secant degeneracy index of the standard strata in the space of binary forms

The space $Pol_d\simeq \bC P^d$ of all complex-valued binary forms of degree $d$ (considered up to a constant factor) has a standard stratification, each stratum of which contains all forms whose set of multiplicities of their distinct roots is given by a fixed partition $\mu \vdash d$. For each such stratum $S_\mu,$ we introduce its secant degeneracy index $\ell_\mu$ which is the minimal number of projectively dependent pairwise distinct points on $S_\mu$, i.e., points whose projective span has dimension smaller than $\ell_\mu-1$. In what follows, we discuss the secant degeneracy index $\ell_\mu$ and the secant degeneracy index $\ell_{\bar \mu}$ of the closure $\bar S_\mu$.