= Solution
For a constant-coefficient <linear multistep method>, <zero-stability> means uniform boundedness of the propagation of starting perturbations as $h\to0$. Equivalently, every solution of the homogeneous <linear recurrence>
$$
\sum_{j=0}^k\alpha_jv_{n+j}=0
$$
satisfies $|v_n|\leq K\max_{0\leq j<k}|v_j|$ for all $n\geq0$, with $K$ independent of $n$ and the starting values. The equivalent <root condition for a multistep method> is that every <root of a polynomial> $\zeta$ of the degree-$k$ <polynomial> $\rho$ satisfies $|\zeta|\leq1$, and every root on $|\zeta|=1$ is simple. Indeed, a root of multiplicity $m$ produces recurrence modes $n^q\zeta^n$, $0\leq q<m$: repeated roots strictly inside the disk are harmless, whereas roots outside it or repeated roots on its boundary produce unbounded modes.
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