A numerical-symbolic algorithm for computing the multiplicity of a component of an algebraic set

Let F1, F2, . . . , Ft be multivariate polynomials (with complex coefficients) in the variables z1, z2, . . . , zn. The common zero locus of these polynomials, V (F1, F2, . . . , Ft) = {p ∈ C|Fi(p) = 0 for 1 ≤ i ≤ t}, determines an algebraic set. This algebraic set decomposes into a union of simpler, irreducible components. The set of polynomials imposes on each component a positive integer known as the multiplicity of the component. Multiplicity plays an important role in many practical applications. It determines roughly “how many times the component should be counted in a computation.” Unfortunately, many numerical methods have difficulty in problems where the multiplicity of a component is greater than one. The main goal of this paper is to present an algorithm for determining the multiplicity of a component of an algebraic set. The method sidesteps the numerical stability issues which have obstructed other approaches by incorporating a combined numerical-symbolic technique.