= Root-multiplicity criterion for stable binary forms
{title2=$[f]\text{ stable}\iff m_p(f)<d/2\text{ for every }p\in\mathbb P^1$}
Under $\operatorname{diag}(t,t^{-1})$, 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.
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