= Exterior roots of the derivative polynomial bound multistep stability
{title2=$\sigma(w_*)=0$, $|w_*|>1$}
If the derivative <characteristic polynomial> $\sigma$ of a <linear multistep method> has a simple root outside the <unit disk> and keeps its degree in the limit, then its <absolute stability> region is bounded. Otherwise a stable sequence with $|z|\to\infty$ would give amplification polynomials $\sigma-\rho/z$ tending coefficientwise to $\sigma$. A root near the exterior root would persist by continuity, contradicting the <root condition for a multistep method>. This argument excludes stability at every sufficiently large complex $z$, not only along the negative real axis.
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