Name
Title | ||
|---|---|---|
A numerical-symbolic algorithm for computing the multiplicity of a component of an algebraic set |
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. |
Year | DOI | Venue |
|---|---|---|
2006 | 10.1016/j.jco.2006.04.003 | J. Complexity |
Keywords | DocType | Volume |
irreducible component,Irreducible components,68W30,multivariate polynomial,Numerical algebraic geometry,Homotopy continuation,Multiplicity,Embedding,numerical stability issue,Generic points,numerical method,important role,Polynomial system,complex coefficient,combined numerical-symbolic technique,main goal,numerical-symbolic algorithm,Primary decomposition,algebraic set,65H10,algebraic set decomposes | Journal | 22 |
Issue | ISSN | Citations |
4 | Journal of Complexity | 16 |
PageRank | References | Authors |
1.11 | 6 | 3 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Dan Bates | 1 | 16 | 1.11 |
| Chris Peterson | 2 | 68 | 10.93 |
| Andrew J. Sommese | 3 | 412 | 39.68 |