2022/12/24 by Hollie Baker, Baker, Hollie
Computer Science · Engineering · #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Computational Geometry and Mesh Generation #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2212.12747
openalex publication_date 2022/12/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A Cylindrical Algebraic Decomposition (CAD) is a decomposition of Rn 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.