A Note on the Multiplicity of the Distinguished Points

Let P(x) be a system of polynomials in s variables, where x∈Cs. If z0 is an isolated zero of P, then the multiplicity and its structure at z0 can be revealed by the normal set of the quotient ring R() or its dual space R* or by certain numerical methods. In his book titled “Numerical Polynomial Algebra”, Stetter described the so-called distinguished points, which are embedded in a zero manifold of P, and the author defined their multiplicities. In this note, we will generalize the definition of distinguished points and give a more appropriate definition for their multiplicity, as well as show how to calculate the multiplicity of these points.