Kossivi Adjamagbo

A resultant criterion and formula for the inversion of a rational map in two variables

Abstract A criterion is given for the invertibility of a polynomial map in two variables over an arbitrary field. A formula for the inverse is derived as wel as other results obtained by McKay and Wang.

A differential criterion and formula for the inversion of a polynomial map in several variables

is invertible (with a polynomial inverse). This conjecture is still open for all n 2 2! Many attempts have been made to prove this conjecture and several partial results have been obtained (see the nice survey paper [l] for more information concerning the Jacobian conjecture). If the Jacobian conjecture is true, it gives a criterion to describe if a polynomial map is invertible. It is the aim of this paper to give a new criterion for the inverti- bility of a polynomial map. Furthermore, in contrast with the Jacobian criterion, our criterion also gives an explicit formula for the inverse. 2. Preliminaries In this section we collect some useful facts concerning polynomial maps. 2.1. Let

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

It is usually admitted, even by the specialists of the Jacobian Conjecture, that it has no hope to be correctly formulated over fields of positive characteristic. This opinion is based on the well known counter-example F = X − X P of a polynomial in one indeterminate X over the prime field F p of cardinality p > 0, whose derivate is 1 and who does not define an automorphism of the F p -algebra F p [X]. But we could remark that the geometric degree of F, i.e. the dimension of the field F p (X) over F p (F), is a multiple of p. From our point of view, this fact is the only accident which could made the traditional formulation of the Jacobian Conjecture fall down in characteristic p. Hence, we think that it is sufficientce to avoid this accident to obtain the right and universal formulation of the classical Jacobian conjecture for the automorphisms of the algebras of polynomials in any number of polynomials over any domain of any characteristic (see its precise statement in 3.1).

A new inversion formula for a polynomial map in two variables

Abstract By computing one resultant, the inverse of an invertible polynomial map in two variables is given.

Eulerian operators and the Jacobian Conjecture, III

Let k be a field of characteristic zero and F:kn→kn a polynomial map with det JFϵk∗ and F(0)=0. Using the Euler operator it is shown that if the k-subalgebra of Mn(k[k1,…,xn]) generated by the homogeneous components of the matrices JF and (JF)-1 is finite-dimensional over k and such that each element in it is a Jacobian matrix, then F is invertible. This implies a result of Connell and Zweibel. Furthermore, it is shown that the Jacobian Conjecture is equivalent with the statement that for every F with det JF ϵ k∗ and F(0)=0, the shifted Euler operator 1+ΣFi(∂∂Fi) is Eulerian.

Refined Noether Normalization Theorem and Sharp Degree Bounds for Dominating Morphisms

Abstract After the proof of a not very known refinement of the Noether Normalization Theorem, we obtain two sharp degree bounds for the geometric degree of a dominating morphism of irreducible affine algebraic varieties and for the degree of the components of the inverse of an isomorphism of such varieties, generalizing by this way the well-known Gabber bound for automorphisms of affine spaces.