= Solution
Insert an exact smooth solution into the <linear multistep method> and define its unscaled <local truncation error> by
$$
\mathcal R_h[y]=\sum_{\ell=0}^s\rho_\ell y(t+\ell h)-h\sum_{\ell=0}^s\sigma_\ell y'(t+\ell h).
$$
Its <Taylor expansion> has coefficients
$$
\mathcal R_h[y]=c_0y(t)+\sum_{j\geq1}c_jh^jy^{(j)}(t),\qquad
c_0=\sum_\ell\rho_\ell,\quad c_j=\frac1{j!}\sum_\ell\ell^j\rho_\ell-\frac1{(j-1)!}\sum_\ell\ell^{j-1}\sigma_\ell.
$$
The method has order at least $p$ exactly when $c_0=\cdots=c_p=0$, since then $\mathcal R_h[y]=O(h^{p+1})$ for every sufficiently smooth $y$. Necessity can be checked by inserting polynomials of successive degrees. This also corresponds to the usual truncation error $\mathcal R_h/h=O(h^p)$.
On the other hand, expansion of the <characteristic polynomials of a linear multistep method> gives
$$
\rho(e^z)-z\sigma(e^z)=c_0+c_1z+c_2z^2+\cdots.
$$
Thus the <exponential-symbol order criterion for a multistep method> is
$$
\boxed{\rho(e^z)-z\sigma(e^z)=O(z^{p+1})\ \Longleftrightarrow\ \text{order at least }p.}
$$
If “order $p$” is meant to be the exact rather than guaranteed order, one additionally requires $c_{p+1}\ne0$. The first two vanishing conditions are the usual <consistency of a numerical method> relations $\rho(1)=0$, $\rho'(1)=\sigma(1)$.
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