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 ∈ Cn|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.

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