Regular chains and triangular decompositions are fundamental and well-developed tools for describing the complex solutions of polynomial systems. This paper proposes 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 regular semi-algebraic systems. We propose two specifications (full and lazy) of such a decomposition and present corresponding algorithms. Under some simplifying assumptions, the lazy decomposition can be computed in singly exponential time w.r.t. the number of variables. We have implemented our algorithms and present experimental results illustrating their effectiveness.
Chen, C., Davenport, J. H., May, J. P., Moreno Maza, M., Xia, B., & Xiao, R. (2013). Triangular decomposition of semi-algebraic systems. Journal of Symbolic Computation, 49, 3-26. https://doi.org/10.1016/j.jsc.2011.12.014