  We construct Nicolai maps for supersymmetric Yang-Mills theory in four and
ten spacetime dimensions in the light-cone gauge, where the elimination of
non-propagating degrees of freedom causes nonlocal and four-fermi interactions
in the Lagrangian. The presence of the latter used to be an obstruction to the
Nicolai map, which has recently been overcome at the price of quantum
corrections to the map. No gauge-fixing or ghost terms arise in this
formulation, since only physical transverse degrees of freedom occur. We
present an explicit form of the Nicolai map to second order in the gauge
coupling. In four dimensions, a `chiral' choice of the map leaves one of the
two transverse gauge-field modes invariant, which forces the classical part of
the map (on the other mode) to become a polynomial (quadratic in the gauge
coupling, cubic in the gauge field)! In the power series expansion for the
ten-dimensional map however, cancellations at each order in the coupling are
systematic but incomplete, still leaving an infinite power series for the
Nicolai map (on all eight transverse modes). Nevertheless, the existence of a
polynomial variant is conceivable, also for the maximal ${\cal N}{=}\,4$ theory
in four dimensions.
