 Root-multiplicity criterion for stable binary forms

Under $\mathrm{diag}(t,t^{-1})$, acting by inverse substitution, $X^{d-j}{Y}^j$ has weight $2j-d$. If $r$ is the smallest active exponent of $Y$, the multiplicity at $[1:0]$ is $r$ and the Hilbert-Mumford value is $d-2r$. Conjugating replaces this point by any projective point. Thus strict positivity is exactly the displayed condition; replacing the strict inequality by $m_p\le d/2$ gives semistability. A binary quartic is stable precisely when it has no repeated projective root.