Collectanea Mathematica

, Volume 68, Issue 3, pp 419–432 | Cite as

Local Bézout theorem for Henselian rings



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 \(\mathfrak {m}\)-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.


Local Bézout Theorem Henselian rings Roots continuity Stable computations Constructive Algebra 

Mathematics Subject Classification

Primary 13J15 Secondary 13P10 13P15 14Q20 03F65 


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Copyright information

© Universitat de Barcelona 2016

Authors and Affiliations

  1. 1.Universidad ComplutenseMadridSpain
  2. 2.University de Franche-ComtéBesançon CedexFrance

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