Cylindrical Algebraic Decomposition With Frontier Condition

TL;DR

This paper introduces an algorithm for constructing Cylindrical Algebraic Decompositions (CADs) that satisfy the frontier condition, enhancing the analysis of semialgebraic sets with applications in topology, motion planning, and triangulation.

Contribution

It proves the existence of a CAD with the frontier condition and presents an elementary complexity algorithm to construct such CADs without coordinate changes.

Findings

Algorithm constructs CADs with frontier condition efficiently

Provides an upper bound on the number of cells in such CADs

Uses a recursive approach based on lexicographical cell indices

Abstract

A Cylindrical Algebraic Decomposition (CAD) is a decomposition of R^n into a finite collection of semialgebraic cells. A CAD satisfies the "frontier condition" if, for every cell C, there is a collection of cells of the decomposition whose union is the closure of C. This property is referred to in other literature as "closure finiteness" or "boundary coherence". This paper proves the existence of, and presents an algorithm to construct, a CAD satisfying the frontier condition without a preliminary change of coordinates, e.g., in the potential presence of blow-ups. The algorithm has elementary (in the sense of L. Kalmar) complexity. This also provides an upper bound on the number of cells in a CAD with this property. The frontier condition can be useful in computing topological properties of semialgebraic sets defined by first-order formulas, in solving motion planning problems and in triangulations of definable monotone families. The algorithm presented takes a novel approach in that it uses a recursion on the lexicographical order of cell indices in the initial decomposition.