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##
Local Bézout Theorem for Henselian rings.

Le papier sur arXiv.

with Mari-Emi Alonso.
Collectanea Matematica (2016).
** Abstract :**

This paper gives an elementary proof
of an improved version of
the algebraic Local Bézout Theorem (given by the authors in JSC 45 (2010) p. 975-985).

We remove some ad hoc hypotheses and obtain an optimal algebraic version of the theorem.
Given a system of n polynomials in n indeterminates with coefficients in a local normal domain
(A,m,k) with an algebraically closed quotient field,
which residually defines an isolated point in k^n of multiplicity r , we prove that there are
finitely many zeroes of the system
above the residual zero (i.e., with coordinates in m ), and the sum of their multiplicities is r .

Our proof is based on the border basis technique of computational algebra.

Here we state and prove an *algebraic version*
of this theorem in the setting of arbitrary Henselian rings and m-adic topology. We are somehow inspired by Arnold, exploiting an abstract version of Weierstrass division (in a Henselian ring) and we introduce also an abstract version of which he called the "multilocal ring". Roughly speaking we consider a
finitely presented A-algebra, where (A,m,k) is a local ring such that the special point is a k-algebra with an isolated zero of multiplicity r and we prove that the "multilocal ring"
determined by this point is a free A-module of rank r .