  The $\mathit{growth\ rate\ function}$ for a nonempty minor-closed class of
matroids $\mathcal{M}$ is the function $h_{\mathcal{M}}(n)$ whose value at an
integer $n \ge 0$ is defined to be the maximum number of elements in a simple
matroid in $\mathcal{M}$ of rank at most $n$. Geelen, Kabell, Kung and Whittle
showed that, whenever $h_{\mathcal{M}}(2)$ is finite, the function
$h_{\mathcal{M}}$ grows linearly, quadratically or exponentially in $n$ (with
base equal to a prime power $q$), up to a constant factor.
  We prove that in the exponential case, there are nonnegative integers $k$ and
$d \le \tfrac{q^{2k}-1}{q-1}$ such that $h_{\mathcal{M}}(n) =
\frac{q^{n+k}-1}{q-1} - qd$ for all sufficiently large $n$, and we characterise
which matroids attain the growth rate function for large $n$. We also show that
if $\mathcal{M}$ is specified in a certain `natural' way (by intersections of
classes of matroids representable over different finite fields and/or by
excluding a finite set of minors), then the constants $k$ and $d$, as well as
the point that `sufficiently large' begins to apply to $n$, can be determined
by a finite computation.
