= Solution
By the <causality root criterion for an autoregressive model>, choose $r>1$ smaller than the modulus of every root of $\Phi$. The function $\Psi=1/\Phi$ is an <analytic function> on and inside $|z|=r$. Writing $M=\max_{|z|=r}|1/\Phi(z)|$, the <Cauchy estimate> gives $|\psi_j|\leq Mr^{-j}$. For $h\geq0$, <independence> of the noise in the <infinite moving-average representation> gives
$$
\gamma(h)=\sigma^2\sum_{j\geq0}\psi_j\psi_{j+h},\qquad
|\gamma(h)|\leq\frac{\sigma^2M^2}{1-r^{-2}}r^{-h}.
$$
The <autocovariance> is symmetric in the lag. Thus \b[exponential decay holds] with
$$
\boxed{s=r^{-1}\in(0,1),\qquad C=\frac{\sigma^2M^2}{1-r^{-2}}.}
$$
This is <exponential autocovariance decay of a causal autoregression>. If $\Phi$ is constant, there are no roots and any $r>1$ works.
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