Triangular Decomposition of Semi-algebraic Systems

TL;DR

This paper extends the concept of triangular decomposition from polynomial systems to semi-algebraic systems, providing algorithms for their decomposition into regular semi-algebraic systems with efficient computation and practical implementation.

Contribution

It introduces novel algorithms for decomposing semi-algebraic systems into regular semi-algebraic systems, adapting tools from polynomial system solving to the real case.

Findings

Decomposition into finitely many regular semi-algebraic systems achieved

Algorithms can be computed in singly exponential time under certain conditions

Experimental results demonstrate the effectiveness of the proposed methods

Abstract

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 {\em 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 w.r.t.\ the number of variables. We implement our algorithms and the experimental results illustrate their effectiveness.