Local Bezout Theorem

fichier pdf.
with Mari-Emi Alonso.
Journal of Symbolic Computation 45, (2010), 975-985.

Abstract :
We give an elementary proof of what we call Local Bezout Theorem. Given a system of  n  polynomials in  n  indeterminates with coefficients in a henselian local domain,  (V,m,k) , which residually defines an isolated point in  kn  of multiplicity  r , we prove (under some additional hypothesis on  V ) 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 techniques of computational algebra.