On Separable Algebras over a U.F.D. and the Jacobian Conjecture in Any Characteristic

Abstract

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).

About Dixmier's conjecture

“…It is unknown whether α − JC 2 implies α − D 1 , as was mentioned there: "In the previous section we proved the starred Dixmier conjecture (in dimension 1) and in this section we will prove the starred Jacobian conjecture in dimension two, however we do not know if one can deduce the former directly from the latter as in the star-free case". Remark 1.1.…”

The complete solution to Bass Generalized Jacobian Conjecture

“…unramified and flat) morphism F from X to Y , atleast when the varieties X, Y, Z are irreducible and non singular and Z is closed in X? According to [4], Lemma 3.4 and [1], Theorem 3, this question in the special case where X = Z is the complex affine space of dimension n is equivalent to the Classical Jacobian Conjecture in dimension n. Unfortunately, Corollary 1 below brings a negative answer to this question in general when X and Y are assumed only to be irreducible and non singular. But the question remains open in the mentioned special case !…”

“…The Classical Jacobian Conjecture claims that any unramified endomorphism of a complex affine space is an automorphism, which means in more ordinary terms that for any integer n > 0, any polynomial map from C n to itself with an invertible jacobian function is itself invertible and its inverse is again a polynomial map (see for instance [7] and [1] or [3] for the 'right" version of this conjecture in any characteristic). From this last point of view, this conjecture may be considered as a global version for polynomial maps of the classical Local Inversion Theorem, which explains largely the fascination that this conjecture exerts on generations of searchers for more than half a century.…”

An approach to the Jacobian Conjecture in terms of irreducibility and square-freeness

“…A positive solution of this problem was obtained by Jelonek ([17], [18]). In 2006 Bakalarski proved an analogical fact for irreducible polynomials over C ( [5], Theorem 3.7, see also [1]). Namely, he proved that a complex polynomial endomorphism is an automorphism if and only if it maps irreducible polynomials to irreducible polynomials.…”