The real Jacobian conjecture, also called the strong real Jacobian conjecture, asserts that a polynomial map 
with nowhere-zero Jacobian determinant is an injection.
Pinchuk (1994) disproved it in dimension two by constructing a noninjective polynomial
map whose Jacobian determinant
is everywhere positive. Adding identity coordinates gives counterexamples in every
dimension 
.
The nowhere-zero determinant in this statement need not be constant. This distinguishes the real conjecture from the constant-determinant condition in the Jacobian conjecture.
For polynomial maps satisfying additional conditions, Hu et al. (2026) proved a sufficient criterion for injectivity. Suppose that 
and 
for all 
. Choose positive integer weights for which all components
of 
have the same weighted degree, and let 
consist of their highest weighted
homogeneous parts. If 
for every 
, then 
is an injection (Hu et al.
2026).
." 23 Oct 2026.