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

Authors:
Dan Bates;Chris Peterson;Andrew J. Sommese
Affiliations:
Department of Mathematics, University of Notre Dame, Notre Dame, IN;Department of Mathematics, Colorado State University, Fort Collins, CO;Department of Mathematics, University of Notre Dame, Notre Dame, IN
Venue:
Journal of Complexity
Year:
2006

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Abstract

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.