= Root-multiplicity criterion for unstable binary forms
{title2=$f\text{ unstable}\iff f=0\text{ or a root has multiplicity }>d/2$}
For $\operatorname{diag}(t,t^{-1})$, the monomial $x^{d-i}y^i$ has weight $d-2i$. Positive weights force divisibility by $x^{\lfloor d/2\rfloor+1}$. Conjugating the subgroup replaces $x$ by any nonzero linear factor. Thus a nonzero <binary form> is unstable exactly when a projective root has multiplicity greater than half its degree. Equality gives zero weight and does not imply instability.
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