On some Exotic Cylindrical Algebraic Decompositions and Cells

TL;DR

This work analyzes high-dimensional topological properties of Cylindrical Algebraic Decompositions (CADs), focusing on closure-finiteness and well-borderedness. By building explicit 4D CADs from 3D bases and employing a cornet cell that is equiregular to a slit disk, the authors construct counterexamples to longstanding conjectures, including CF versus WB equivalence in $^4$, fibre structure nonconformity, and non-regular sectors with regular bounds. The results introduce exotic CAD cells and the equiregularity framework to transfer boundary topology from lower-dimensional bases to higher-dimensional sectors and sections. These findings reveal limitations in assuming certain topological properties in higher dimensions, with implications for CAD-based algorithms and the understanding of semi-algebraic set topology.

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.

Figures (4)

• Figure 1: The cornet cell $\mathfrak{C}$ and the the slit disk $\mathbb{D}_{\text{s}}$. Their boundary are in dashed lines.
• Figure 2: The the section $\mathbb{D}_{\text{s}} \odot \{f\}$ (blue), its boundary (thick black) and of the fibre $(\pi|_{\overline{\mathbb{D}_{\text{s}} \odot \{f\}}})^{-1}(\{\varphi(\textbf{x})\})$ (red dots) of Proposition \ref{['prop:fibre']}.
• Figure 3: Two different views of the section $\mathbb{D}_{\text{s}} \odot \{f\}$ (blue) and its boundary (thick black) of Proposition \ref{['prop:closureContractible']}.
• Figure 4: The sections $\mathbb{D}_{\text{s}} \odot \{-\widetilde{u}\}$ (blue) and $\mathbb{D}_{\text{s}} \odot \{\widetilde{u}\}$ (orange), their boundaries (thick black), and the the sector $\mathbb{D}_{\text{s}} \odot (-\widetilde{u},\widetilde{u})$ (gray) of Proposition \ref{['prop:conj722']}. Note that the sector in red does not meet $\mathbb{D}_{\text{s}} \odot (-\widetilde{u},\widetilde{u})$ but is a subset of its closure.

Theorems & Definitions (20)

• definition 1
• definition 2
• Theorem 2.1
• Lemma 2.2
• proof
• proof : of Theorem \ref{['thrm:CFnotWB']}
• remark 1
• definition 3
• definition 4
• Proposition 3.1