The growthΒ rateΒ function for a nonempty minor-closed class of
matroids M is the function hMβ(n) whose value at an
integer nβ₯0 is defined to be the maximum number of elements in a simple
matroid in M of rank at most n. Geelen, Kabell, Kung and Whittle
showed that, whenever hMβ(2) is finite, the function
hMβ 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
dβ€qβ1q2kβ1β such that hMβ(n)=qβ1qn+kβ1ββqd for all sufficiently large n, and we characterise
which matroids attain the growth rate function for large n. We also show that
if 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