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.