On system of parameters, local intersection multiplicity and Bezout’s theorem

Abstract

This paper provides effective methods for computing the local intersection multiplicity as the length of a well-defined ideal (see Theorem and Proposition 1). There are other ways of obtaining such an ideal (see [2], [9], [12], [18]) but ours is simpler because of our use of reducing systems of parameters. Applying these ideal theoretic methods we will give a new and simple proof of Bezout’s Theorem (see §4). Hence this proof again provides the connection between the different viewpoints which are treated in the work of Lasker-Macaulay-Gröbner and Severi-van der Waerden-Weil concerning the multiplicity theory.