Abstract

Abstract. 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. Keywords. Symbolic-numeric computation, triangular decomposition, singular value decomposition, linear system. 1. Introduction Systems of polynomial equations frequently arise in applications, and it is often ofinterest to determine their triangular forms. Such representations enable the expression of some of the variables as functions of the remaining ("free") variables.Already, methods exist which are designed to compute triangular sets for exact systems whose varieties are of arbitrary dimension (43; 22; 28; 42). However, forreal world problems, the systems under consideration frequently have approximate coefficients that are inferred from experimental data. This means that the stabilityof these triangular representations is a valid concern. To be more clear, we are not merely concerned with the sensitivity of the triangular representation to pertur-bations in the original data, but we also want the solution set to be stable under small changes in values taken by the free variables. This work is supported by NSERC, MITACS, and Maplesoft, Canada.

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