  Every minor-closed class of matroids of bounded branch-width can be
characterized by a list of excluded minors, but unlike graphs, this list may
need to be infinite in general. However, for each fixed finite field $\mathbb
F$, the list needs to contain only finitely many $\mathbb F$-representable
matroids, due to the well-quasi-ordering of $\mathbb F$-representable matroids
of bounded branch-width under taking matroid minors [J. F. Geelen, A. M. H.
Gerards, and G. Whittle (2002)]. But this proof is non-constructive and does
not provide any algorithm for computing these $\mathbb F$-representable
excluded minors in general.
  We consider the class of matroids of path-width at most $k$ for fixed $k$. We
prove that for a finite field $\mathbb F$, every $\mathbb F$-representable
excluded minor for the class of matroids of path-width at most $k$ has at most
$2^{|\mathbb{F}|^{O(k^2)}}$ elements. We can therefore compute, for any integer
$k$ and a fixed finite field $\mathbb F$, the set of $\mathbb F$-representable
excluded minors for the class of matroids of path-width $k$, and this gives as
a corollary a polynomial-time algorithm for checking whether the path-width of
an $\mathbb F$-represented matroid is at most $k$. We also prove that every
excluded pivot-minor for the class of graphs having linear rank-width at most
$k$ has at most $2^{2^{O(k^2)}}$ vertices, which also results in a similar
algorithmic consequence for linear rank-width of graphs.
