  We consider a formulation of gauge field theory where the gauge field
$A_\alpha$ and the field strength $F_{\alpha\beta}$ are independent variables,
as in the Palatini formulation of gravity. For the simplest gauge field action,
this is known to be equivalent to the usual formulation. We add non-minimal
couplings between $F_{\alpha\beta}$ and a scalar field, solve for
$F_{\alpha\beta}$ and insert it back into the action. This leads to modified
gauge field and scalar field terms. We consider slow-roll inflation and show
that because of the modifications to the scalar sector, adding higher order
terms to the inflaton potential does not spoil its flatness, unlike in the
usual case. Instead they make the effective potential closer to quadratic. The
modifications also solve the problem that Higgs inflation in the Palatini
formulation is sensitive to higher order terms.
