  Based on detailed spectral synthesis we carry out quantitative measurements
of the strength and complexity of surface magnetic fields in the four
well-known M-dwarfs GJ 388, GJ 729, GJ 285, and GJ 406 populating the mass
regime around the boundary between partially and fully convective stars. Very
high resolution R=100000, high signal-to-noise (up to 400) near-infrared Stokes
I spectra were obtained with CRIRES at ESO's Very Large Telescope covering
regions of the FeH Wing-Ford transitions at 1mum. The field distributions in
all four stars are characterized by three distinct groups of field components,
the data are neither consistent with a smooth distribution of different field
strengths, nor with one average field strength covering the full star. We find
evidence of a subtle difference in the field distribution of GJ 285 compared to
the other three targets. GJ 285 also has the highest average field of 3.5kG and
the strongest maximum field component of 7-7.5kG. The maximum local field
strengths in our sample seem to be correlated with rotation rate. While the
average field strength is saturated, the maximum local field strengths in our
sample show no evidence for saturation. We find no difference between the field
distributions of partially and fully convective stars. The one star with
evidence for a field distribution different to the other three is the most
active star (i.e. with largest x-ray luminosity and mean surface magnetic
field) rotating relatively fast. A possible explanation is that rotation
determines the distribution of surface magnetic fields, and that local field
strengths grow with rotation even in stars in which the average field is
already saturated.
