= Solution
Order $p$ applied to this particular differential equation forces the local stability <polynomial> to agree with the exact exponential through degree $p$:
$$
P_s(z)=e^z+O(z^{p+1}),\qquad
[z^j]P_s=1/j!\quad(0\le j\le p).
$$
As its degree is at most $s$, the nonzero coefficient $1/p!$ cannot be present for $p>s$. Therefore
$$
\boxed{p\le s,\qquad
p=s\ \Longrightarrow\ P_s(z)=\sum_{j=0}^s\frac{z^j}{j!}.}
$$
The implication is a necessary stability-polynomial condition; it does not assert that every such <polynomial> automatically gives the full nonlinear order conditions.
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