Local Bézout Theorem, (2)

This paper gives an elementary proof of an improved version of the algebraic Local B\'ezout Theorem (given by the authors in JSC 45 (2010) 975--985). Here 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 {\em 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 $r$.