Narrow Results

In this paper, using an effective algorithm, we obtain an equivalent statement to the Jacobian Conjecture. For a polynomial map $F$ on an affine space of dimension $n$ over a field of characteristic $0$, we define recursively a finite sequence of polynomial maps. We give an equivalent condition to the invertibility of $F$ as well as a formula for $F^{-1}$ in terms of this finite sequence of polynomial maps. Some examples illustrate the effective aspects of our approach.

The class of Pascal finite polynomial automorphisms is a subclass of the class of locally finite ones allowing a more effective approach. In characteristic zero, a Pascal finite automorphism is the exponential map of a locally nilpotent derivation. However, Pascal finite automorphisms are defined in any characteristic, and therefore constitute a generalization of exponential automorphisms to positive characteristic. In this paper, we prove several properties of Pascal finite automorphisms. We obtain in particular that the Pascal finite property is stable under taking powers but not under composition. This leads us to formulate a generalization of the exponential generators conjecture to arbitrary characteristic.