  Implementations of known reductions of the Strong Real Jacobian Conjecture
(SRJC), to the case of an identity map plus cubic homogeneous or cubic linear
terms, and to the case of gradient maps, are shown to preserve significant
algebraic and geometric properties of the maps involved. That permits the
separate formulation and reduction, though not so far the solution, of the SRJC
for classes of nonsingular polynomial endomorphisms of real n-space that
exclude the Pinchuk counterexamples to the SRJC, for instance those that induce
rational function field extensions of a given fixed odd degree.
