A Numerical Local Dimension Test for Points on the Solution Set of a System of Polynomial Equations

Abstract

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 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 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.

Mechanism mobility and a local dimension test

“…The software package Bertini [2], which solves systems of polynomial equations using continuation, uses a local dimension test [1] in culling out unwanted points that land on positive dimensional sets. We have adapted that test for the current purpose; the result is an algorithm, LocalDimFinder, available at the Bertini website [2].…”

“…A more manageable local dimension test, presented in [1] follows from Proposition 5.1, item 3 and the following observation.…”

“…In [1], this is provided by foreknowledge of a bound, sayμ, on the multiplicity of z. As soon as the sequence exceedsμ, stabilization cannot occur and computation can stop.…”

“…As soon as the sequence exceedsμ, stabilization cannot occur and computation can stop. In the context of numerical continuation algorithms considered in [1], the system f (x) is polynomial, and the multiplicity bound is a count of the number of paths that approach z in a polynomial homotopy.…”

“…This article describes an extension of the Jacobian matrix, called a Macaulay matrix, which includes higher order terms that can be used to distinguish between finite and infinitesimal directions, thus arriving at the finite mobility of the mechanism. The methodology involved comes from work in numerical algebraic geometry, where the Macaulay matrix is central to a local dimension test that is used to sort solution points found by numerical continuation [1].…”