2006/08/01 by Kossivi Adjamagbo, Adjamagbo, Kossivi
Mathematics · #13F15 #14R15 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #math.AG #msc:13F15 #msc:14R15
paper · pdf · doi:10.48550/arxiv.math/0608008
This paper presents the first reformulations in positive characteristic of the Jacobian conjecture in zero characteristic, thanks to methods of model theory. Such reformulations are usefull for the proof of the equivalence of Jacobian, Dixmier and Poisson conjectures in any characteristic, as explained in another ArXiv preprint
The main theorem (2.2) consists in two characterizations of isomorphisms of factorial domains in terms of prime or primary rings elements, and unramified, flat or weakly injective affine schemes morphisms. In order to apply this theorem to the famous Jacobian Conjecture, we first introduce its different versions in any characteristic (3.1), and give two reformulations of some these versions in terms of domains of positive characteristic (3.8) and finite prime fields (3.9). Finally, we deduce from the main theorem an original reformulation of the any characteristic version of the Jacobian Conjecture in terms of prime or primary rings elements (3.11).