Reversible polynomials

Abstract

In this paper we consider a one-dimensional analogue of Jacobian Conjecture. Here polynomials are considered over commutative ring. We have shown that the polynomial is reversible with respect to superposition if and only if its derivative is reversible in the ring of polynomials with respect to usual operation. Usually Jacobian Conjecture is considered over a field. When the characteristic is positive, there exist a counter example to Jacobian Conjecture. Over the field of zero characteristic, analogue of Jacobian Conjecture is solved easily. In this case Jacobian Conjecture holds for polynomials of the first order, to be more precise, x coefficient must be different from zero, but other non-free coefficients must be equal to zero. Currently Jacobian Conjecture for polynomials of two and more variables is still open. The main result is that we have reduced the Jacobian Conjecture to the Jacobian Conjecture for polynomials with components of linear or third degree. For Keller polynomial mappings Jacobian Problem is reduced to Injection of polynomial mappings. In this case in order to solve the Jacobian Conjecture, the main field is required to be algebraically closed.