Computing Cylindrical Algebraic Decomposition via Triangular Decomposition

TL;DR

This paper introduces a novel method that uses comprehensive triangular decomposition to compute cylindrical algebraic decompositions, enhancing the process of analyzing semi-algebraic sets.

Contribution

The paper presents a new approach combining triangular decomposition with cylindrical algebraic decomposition, providing an alternative computational method.

Findings

Implemented the new approach successfully.

Demonstrated effectiveness on example problems.

Potential for improved computational efficiency.

Abstract

Cylindrical algebraic decomposition is one of the most important tools for computing with semi-algebraic sets, while triangular decomposition is among the most important approaches for manipulating constructible sets. In this paper, for an arbitrary finite set $F \subset {\R}[y_1, ..., y_n]$ we apply comprehensive triangular decomposition in order to obtain an $F$-invariant cylindrical decomposition of the $n$-dimensional complex space, from which we extract an $F$-invariant cylindrical algebraic decomposition of the $n$-dimensional real space. We report on an implementation of this new approach for constructing cylindrical algebraic decompositions.