Root-multiplicity criterion for unstable binary forms ($f\text{ unstable}⟺f=0\text{ or a root has multiplicity }>d/2$)

For $\mathrm{diag}(t,t^{-1})$, 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.