  Frick and Grohe [J. ACM 48 (2006), 1184-1206] introduced a notion of graph
classes with locally bounded tree-width and established that every first order
logic property can be decided in almost linear time in such a graph class.
Here, we introduce an analogous notion for matroids (locally bounded
branch-width) and show the existence of a fixed parameter algorithm for first
order logic properties in classes of regular matroids with locally bounded
branch-width. To obtain this result, we show that the problem of deciding the
existence of a circuit of length at most k containing two given elements is
fixed parameter tractable for regular matroids.
