A numerical-symbolic algorithm for computing the multiplicity of a component of an algebraic set (original) (raw)

Let F1, F2, . . . , Ft be multivariate polynomials (with complex coefficients) in the variables z 1 , z 2 , . . . , z n . The common zero locus of these polynomials, V (F 1 , F 2 , . . . , F t ) = {p ∈ C n |F i (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.