  The discriminant of a polynomial map is central to problems from affine
geometry and singularity theory. Standard methods for characterizing it rely on
elimination techniques that can often be ineffective. This paper concerns
polynomial maps on the two-dimensional torus defined over a field of Puiseux
series. We present a combinatorial procedure for computing the tropical curve
of the discriminant of maps determined by generic polynomials with given
supports. Our results enable one to compute the Newton polytope of the
discriminant of complex polynomial maps on the plane.
