Name
Title | ||
|---|---|---|
A Numerical Local Dimension Test for Points on the Solution Set of a System of Polynomial Equations |
Abstract | ||
|---|---|---|
The solution set $V$ of a polynomial system, i.e., the set of common zeroes of a set of multivariate polynomials with complex coefficients, may contain several components, e.g., points, curves, surfaces, etc. Each component has attached to it a number of quantities, one of which is its dimension. Given a numerical approximation to a point $\mathbf{p}$ on the set $V$, this article presents an efficient algorithm to compute the maximum dimension of the irreducible components of $V$ which pass through $\mathbf{p}$, i.e., a local dimension test. Such a test is a crucial element in the homotopy-based numerical irreducible decomposition algorithms of Sommese, Verschelde, and Wampler. This article presents computational evidence to illustrate that the use of this new algorithm greatly reduces the cost of so-called “junk-point filtering,” previously a significant bottleneck in the computation of a numerical irreducible decomposition. For moderate size examples, this results in well over an order of magnitude improvement in the computation of a numerical irreducible decomposition. As the computation of a numerical irreducible decomposition is a fundamental backbone operation, gains in efficiency in the irreducible decomposition algorithm carry over to the many computations which require this decomposition as an initial step. Another feature of a local dimension test is that one can now compute the irreducible components in a prescribed dimension without first computing the numerical irreducible decomposition of all higher dimensions. For example, one may compute the isolated solutions of a polynomial system without having to carry out the full numerical irreducible decomposition. |
Year | DOI | Venue |
|---|---|---|
2009 | 10.1137/08073264X | SIAM J. Numerical Analysis |
Keywords | Field | DocType |
irreducible component,numerical algebraic geometry,solution set,full numerical irreducible decomposition,irreducible decomposition,maximum dimension,mul- tiplicity,. local dimension,numerical irreducible decomposition,homotopy continuation,local dimension test,polynomial system,14q99,generic points,homotopy-based numerical irreducible decomposition,polynomial equations,irreducible components,polynomial system ams subject classification. 65h10,numerical approximation,68w30,numerical local dimension test,higher dimension,multiplicity,generic point | Mathematical optimization,Irreducible component,Polynomial,Mathematical analysis,Multiplicity (mathematics),System of polynomial equations,Decomposition method (constraint satisfaction),Solution set,Irreducible element,Irreducible polynomial,Mathematics | Journal |
Volume | Issue | ISSN |
47 | 5 | 0036-1429 |
Citations | PageRank | References |
15 | 0.86 | 14 |
Authors | ||
4 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Daniel J. Bates | 1 | 103 | 12.03 |
| Jonathan D. Hauenstein | 2 | 269 | 37.65 |
| Chris Peterson | 3 | 68 | 10.93 |
| Andrew J. Sommese | 4 | 412 | 39.68 |