Abstract: Regular chains and triangular decompositions are fundamental and well-developed tools for describing the complex solutions of polynomial systems. We propose adaptations of these tools focusing on solutions of the real analogue: semi-algebraic systems.

We show that any such system can be decomposed into finitely many so-called "regular semi-algebraic systems". We propose two specifications of such a decomposition and present corresponding algorithms. Under some assumptions, one type of decomposition can be computed in singly exponential time with respect to the number of variables. We have implemented our algorithms and the experimental results illustrate their effectiveness. A software demonstration will conclude this talk.

This is a joint work with Changbo Chen (UWO), James H. Davenport (Bath U.), John P. May (Maplesoft), Bican Xia (Peking U.) and Xiao Rong (UWO).

The corresponding article is published in the Proceedings of the 2010 International Symposium of Symbolic and Algebraic Computation (ISSAC'10) and available at ww.csd.uwo.ca/~moreno/Publications/118_paper.pdf