Local Bézout Theorem for Henselian rings

Abstract : In this paper we prove what we call Local Bézout Theorem (Theorem 3.7). It is a formal abstract algebraic version, in the frame of Henselian rings and m-adic topology, of a well known theorem in the analytic complex case. This classical theorem says that, given an isolated point of multiplicity r as a zero of a local complete intersection, after deforming the coefficients of these equations we find in a sufficiently small neighborhood of this point exactly r isolated zeroes counted with multiplicities. Our main tools are, first the border bases [11], which turned out to be an efficient computational tool to deal with deformations of algebras. Second we use an important result of de Smit and Lenstra [7], for which there exists a constructive proof in [13]. Using these tools we find a very simple proof of our theorem, which seems new in the classical literature.
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Collectanea Mathematica, Springer Verlag, 2017, 68 (3), pp.419-432. 〈10.1007/s13348-016-0184-0〉
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Contributeur : Henri Lombardi <>
Soumis le : mercredi 6 décembre 2017 - 18:50:46
Dernière modification le : jeudi 7 décembre 2017 - 01:11:58

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M. Emilia Alonso, Henri Lombardi. Local Bézout Theorem for Henselian rings. Collectanea Mathematica, Springer Verlag, 2017, 68 (3), pp.419-432. 〈10.1007/s13348-016-0184-0〉. 〈hal-01657533〉

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