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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, eachstratum of which contains all forms whose set of multiplicities of theirdistinct roots is given by a fixed partition $\mu \vdash d$. For each suchstratum $S_\mu,$ we introduce its secant degeneracy index $\ell_\mu$ which isthe 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$\bar S_\mu$.

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