= Solution
For an explicit <Runge-Kutta method>, write its strictly lower-triangular stage matrix as $A$, weight vector as $b$, and $\mathbf1=(1,\ldots,1)^T$. Applied to the scalar test equation, its stage vector satisfies
$$
k=\lambda y_n\mathbf1+\lambda hAk.
$$
Since $A^s=0$, inversion is a finite <Neumann series>. Putting $z=\lambda h$ gives
$$
\boxed{y_{n+1}=P_s(z)y_n,\qquad
P_s(z)=1+z\,b^T\sum_{j=0}^{s-1}z^jA^j\mathbf1.}
$$
This proves degree at most $s$. The printed statement of degree exactly $s$ needs a nondegeneracy assumption: a two-stage explicit scheme with $b=(1,0)$ is simply Euler's method with an unused second stage, and has <polynomial> $1+z$ of degree one.
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