p > 0 even for one variable. The characteristic of a field, if it is not zero, must be prime, so at least 2. The polynomial has derivative which is 1 (because px is 0) but it has no inverse function. However, suggested extending the Jacobian conjecture to characteristic by adding the hypothesis that p does not divide the degree of the field extension . The existence of a polynomial inverse is obvious if F is simply a set of functions linear in the variables, because then the inverse will also be a set of linear functions. A very simple non-linear example would be if in which case the Jacobian matrix is whose determinant is the constant 1. We can then write the inverse as the polynomials But if we modify F slightly, as then the determinant is not constant, the Jacobian conjecture does not apply, and in fact we find: The expression for x is not a polynomial. The condition JF ≠ 0 is related to the inverse function theorem in multivariable calculus. In fact for smooth functions (and so in particular for polynomials) a smooth local inverse function to F exists at every point where JF is non-zero. For example, the map x → x + x3 has a smooth global inverse, but the inverse is not polynomial. Results Stuart Sui-Sheng Wang proved the Jacobian conjecture for polynomials of degree 2. Hyman Bass, Edwin Connell, and David Wright showed that the general case follows from the special case where the polynomials are of degree 3, or even more specifically, of cubic homogeneous type, meaning of the form F = (X1 + H1, ..., Xn + Hn), where each Hi is either zero or a homogeneous cubic. Ludwik Drużkowski showed that one may further assume that the map is of cubic linear type, meaning that the nonzero Hi are cubes of homogeneous linear polynomials. It seems that Drużkowski's reduction is one most promising way to go forward. These reductions introduce additional variables and so are not available for fixed N. Edwin Connell and Lou van den Dries proved that if the Jacobian conjecture is false, then it has a counterexample with integer coefficients and Jacobian determinant 1. In consequence, the Jacobian conjecture is true either for all fields of characteristic 0 or for none. For fixed dimension N, it is true if it holds for at least one algebraically