On Approximate Linearized Triangular Decompositions

In this paper, we describe progress on the development of algorithms for triangular decomposition of approximate systems. We begin with the treatment of linear, homogeneous systems with positive-dimensional solution spaces, and approximate coefficients. We use the Singular Value Decomposition to decompose such systems into a stable form, and discuss condition numbers for approximate triangular decompositions. Results from the linear case are used as the foundation of a discussion on the fully nonlinear case. We introduce linearized triangular sets, and show that we can obtain useful stability information about sets corresponding to different variable orderings. Examples are provided, experiments are described, and connections with the works of Sommese, Verschelde, and Wampler are made.