On some Exotic Cylindrical Algebraic Decompositions and Cells
Lucas Michel
TL;DR
This paper constructs explicit examples of cylindrical algebraic decompositions with specific topological properties, challenging existing conjectures and advancing understanding in algebraic geometry and computational topology.
Contribution
It provides explicit counterexamples to several longstanding conjectures about topological properties of CADs, enriching the theoretical landscape.
Findings
Refutes multiple conjectures on CAD topologies
Provides explicit examples of CADs with unusual properties
Advances understanding of CAD topological assumptions
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.
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Polynomial and algebraic computation · Robotic Mechanisms and Dynamics