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Article Dans Une Revue Collectanea Mathematica Année : 2017

Local Bézout Theorem for Henselian rings

Résumé

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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Dates et versions

hal-01657533 , version 1 (06-12-2017)

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