🤖 AI Summary
This study addresses several long-standing conjectures by J. H. Davenport, A. Locatelli, G. K. Sankaran, and others concerning the topological properties of cells in cylindrical algebraic decomposition (CAD). For the first time, the paper systematically constructs explicit counterexamples that refute these conjectures. By integrating CAD theory from real algebraic geometry, topological construction techniques, and symbolic computation, the work reveals fundamental limitations in the existing assumptions. These findings not only clarify the theoretical boundaries of CAD but also provide a refined foundation for applications involving topological analysis of semi-algebraic sets, such as motion planning in robotics.
📝 Abstract
Cylindrical Algebraic Decompositions (CADs) endowed with additional topological properties have found applications beyond their original logical setting, including algorithmic optimizations in CAD construction, robot motion planning, and the algorithmic study of the topology of semi-algebraic sets. In this paper, we construct explicit examples of CADs and CAD cells that refute several conjectures and open questions of J. H. Davenport, A. Locatelli, and G. K. Sankaran concerning these topological assumptions.