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

In: Automorphisms of Affine Spaces

Author

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  • Kossivi Adjamagbo

    (Université de Paris VI, UFR 920, UMR C9994)

Abstract

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).

Suggested Citation

  • Kossivi Adjamagbo, 1995. "On Separable Algebras over a U.F.D. and the Jacobian Conjecture in Any Characteristic," Springer Books, in: Arno van den Essen (ed.), Automorphisms of Affine Spaces, pages 89-103, Springer.
    • Handle: RePEc:spr:sprchp:978-94-015-8555-2_5
    DOI: 10.1007/978-94-015-8555-2_5
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