= Solution
<Zero-stability> means that perturbations of the starting values of the zero-step recurrence remain bounded by a fixed multiple of their initial size, uniformly as the number of steps tends to infinity on a fixed time interval. In particular solutions of $\sum_{j=0}^k\alpha_je_{n+j}=0$ obey $\max_{n\leq N}|e_n|\leq C\max_{j<k}|e_j|$ with $C$ independent of $N$.
The equivalent <root condition for a multistep method> is
$$
\boxed{\text{Every root of }\rho(\zeta)\text{ has }|\zeta|\leq1,\text{ and roots with }|\zeta|=1\text{ are simple}.}
$$
A root outside the unit circle gives exponential growth of a <recurrence relation> mode; a repeated root on it gives polynomial growth. Repeated roots strictly inside it are harmless because their polynomial factor is dominated by decay.
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