Numerical irreducible decomposition using projections from points on the components
Andrew John Sommese, Jan Verschelde, Charles W. Wampler · Contemporary mathematics - American Mathematical Society · 2001
. To classify positive dimensional solution components of a polynomial system, we construct polynomials interpolating points sampled from each component. In previous work, points on an i-dimensional component were linearly projected onto a generically chosen (i + 1)-dimensional subspace. In this paper, we present two improvements. First, we reduce the dimensionality of the ambient space by determining the linear span of the component and restricting to it. Second, if the dimension of the linear span is greater than i + 1, we use a less generic projection that leads to interpolating polynomials of lower degree, thus reducing the number of samples needed. While this more ecient approach still guarantees | with probability one | the correct determination of the degree of each component, the mere evaluation of an interpolating polynomial no longer certies the membership of a point to that component. We present an additional numerical test that certies membership in this new si...