Local B\'ezout Theorem for Henselian rings

TL;DR

This paper provides an elementary, optimal algebraic proof of the Local Bézout Theorem for Henselian rings, removing previous ad hoc assumptions and establishing a clear relationship between multiplicities of solutions in local algebraic systems.

Contribution

It introduces an algebraic version of the Local Bézout Theorem applicable to Henselian rings, using border basis techniques and abstract Weierstrass division, improving and generalizing prior results.

Findings

Finiteness of solutions above an isolated zero with multiplicity r

Sum of multiplicities of solutions equals r

Multilocal ring is a free module of rank r over the base ring

Abstract

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 $r$.