  The dominant component of the (100 MeV - 50 GeV) GRB emission detected by LAT
starts with a delay relative to the prompt soft (sub-MeV) gamma-rays and lasts
long after the soft component fades. This has lead to the intriguing suggestion
that this high energy emission is generated via synchrotron emission of
relativistic electrons accelerated by the external shock. Moreover, the limits
on the MeV afterglow emission lead to the suggestion that, at least in bright
GeV bursts the field is not amplified beyond compression in the shock. We show
here that considerations of confinement (within the decelerating shock),
efficiency and cooling of the emitting electrons constrain, within this model,
the magnetic fields that arise in both the upstream (circum burst) and
downstream (ejecta) regions, allowing us to obtain a direct handle on their
values. The well known limit on the maximal synchrotron emission, when combined
with the blast wave evolution, implies that late photons (arriving more than
~100 s after the burst) with energies higher than ~ 10GeV do not arise
naturally from external shock synchrotron and almost certainly have a different
origin. Finally, even a modest seed flux (a few mJy) at IR-optical would
quench, via Inverse Compton cooling, the GeV emission unless the magnetic field
is significantly amplified behind the shock. An observation of a burst with
simultaneous IR-optical and GeV emission will rule out this model.
