  The lack of any local solution to the first-order-in-h omegamn Seiberg-Witten
(SW) map equations for U(1) vector superfields compels us to obtain the most
general solution to those equations that is a quadratic polynomial in the
ordinary vector superfield, v, its chiral and antichiral projections and the
susy covariant derivatives of them all. Furnished with this solution, which is
local in the susy Landau gauge, we construct an ordinary dual of noncommutative
U(1) SYM in terms of ordinary fields which carry a linear representation of the
N=1 susy algebra. By using the standard SW map for the N=1 U(1) gauge
supermultiplet we define an ordinary U(1) gauge theory which is dual to
noncommutative U(1) SYM in the WZ gauge. We show that the ordinary dual so
obtained is supersymmetric, for, as we prove as we go along, the ordinary gauge
and fermion fields that we use to define it carry a nonlinear representation of
the N=1 susy algebra. We finally show that the two ordinary duals of
noncommutative U(1) SYM introduced above are actually the same N=1 susy gauge
theory. We also show in this paper that the standard SW map is never the theta
theta--bar component of a local superfield in v and check that, at least at a
given approximation, a suitable field redefinition of that map makes the
noncommutative and ordinary --in a Bmn field-- susy U(1) DBI actions
equivalent.
