arXiv subjects

Lucas Michel

Publications and source records attributed to Lucas Michel.

6 recordsLinked to original sources

arXiv2026

On Minimum CADs for Algebraic Sets in Dimension Three

Cylindrical Algebraic Decomposition (CAD) algorithms typically produce a decomposition adapted to a finite family of semi-algebraic sets $\mathcal{F}$ (i.e. every member of $\mathcal{F}$ is a union of cells). Different algorithms may produce different outputs, and introduce unnecessary cell divisions. Recent work by Michel, Mathonet, and Z\'ena\"idi in ISSAC 2024 formalised this issue by studying the refinement order on the set of all CADs adapted to $\mathcal{F}$ and analysing the existence of a minimum (coarsest) adapted CAD. It was shown that such a minimum adapted CAD always exists for subsets of $\mathbb{R}$ and $\mathbb{R}^2$, but not of $\mathbb{R}^n$ ($n \geqslant 3$) in general. It is natural to seek natural classes of subsets of $\mathbb{R}^n$ that admit a minimum adapted CAD. In this paper, we identify a class of subsets of $\mathbb{R}^3$ that contains all algebraic sets for which minimum adapted CADs do exist. This provides the first positive existence theorem for minimum CAD for a non-trivial class of sets.

cs.SC↗

arXiv2026

Further results on Minimal and Minimum Cylindrical Algebraic Decompositions

We consider cylindrical algebraic decompositions (CADs) as a tool for representing semi-algebraic subsets of $\mathbb{R}^n$. In this framework, a CAD $\mathscr{C}$ is adapted to a given set $S$ if $S$ is a union of cells of $\mathscr{C}$. Different algorithms computing an adapted CAD may produce different outputs, usually with redundant cell divisions. In this paper we analyse the possibility to remove the superfluous data. We thus consider the set $\text{CAD}^r(\mathcal{F})$ of CADs of class $C^r$ ($r \in \mathbb{N} \cup \{\infty, \omega\}$) that are adapted to a finite family $\mathcal{F}$ of semi-algebraic sets of $\mathbb{R}^n$, endowed with the refinement partial order and we study the existence of minimal and minimum element in $\text{CAD}^r(\mathcal{F})$. We show that for every such $\mathcal{F}$ and every $\mathscr{C} \in \text{CAD}^r(\mathcal{F})$, there is a minimal CAD of class $C^r$ adapted to $\mathcal{F}$ and smaller (i.e. coarser) than or equal to $\mathscr{C}$. In dimension $n=1$ or $n=2$, this result is strengthened by proving the existence of a minimum element in $\text{CAD}^r(\mathcal{F})$. In contrast, for any $n \geq 3$, we provide explicit examples of semi-algebraic sets whose associated poset of adapted CADs does not admit a minimum. We then introduce a reduction relation on $\text{CAD}^r(\mathcal{F})$ in order to define an algorithm for the computation of minimal CADs and we characterise those semi-algebraic sets $\mathcal{F}$ for which $\text{CAD}^r(\mathcal{F})$ has a minimum by means of confluence of the associated reduction system. We finally provide practical criteria for deciding if a semi-algebraic set does admit a minimum CAD and apply them to describe various concrete examples of semi-algebraic sets, along with their minimum CAD of class $C^r$.

cs.SC↗

arXiv2026

On some Exotic Cylindrical Algebraic Decompositions and Cells

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.

math.AG↗

arXiv2025

Projective Delineability for Single Cell Construction

The cylindrical algebraic decomposition (CAD) is the only complete method used in practice for solving problems like quantifier elimination or SMT solving related to real algebra, despite its doubly exponential complexity. Recent exploration-guided algorithms like NLSAT, NuCAD, and CAlC rely on CAD technology but reduce the computational effort heuristically. Single cell construction is a paradigm that is used in each of these algorithms. The central property on which the CAD algorithm is based is called delineability. Recently, we introduced a weaker notion called projective delineability which can require fewer computations to guarantee, but needs to be applied carefully. This paper adapts the single cell construction for exploiting projective delineability and reports on experimental results.

cs.SC↗

arXiv2024

On Projective Delineability

We consider cylindrical algebraic decomposition (CAD) and the key concept of delineability which underpins CAD theory. We introduce the novel concept of projective delineability which is easier to guarantee computationally. We prove results about this which can allow reduced CAD computations.

math.AG↗

arXiv2024

On Minimal and Minimum Cylindrical Algebraic Decompositions

We consider cylindrical algebraic decompositions (CADs) as a tool for representing semi-algebraic subsets of $\mathbb{R}^n$. In this framework, a CAD $\mathscr{C}$ is adapted to a given set $S$ if $S$ is a union of cells of $\mathscr{C}$. Different algorithms computing an adapted CAD may produce different outputs, usually with redundant cell divisions. In this paper we analyse the possibility to remove the superfluous data. More precisely we consider the set CAD$(S)$ of CADs that are adapted to $S$, endowed with the refinement partial order and we study the existence of minimal and minimum elements in this poset. We show that for every semi-algebraic set $S$ of $\mathbb{R}^n$ and every CAD $\mathscr{C}$ adapted to $S$, there is a minimal CAD adapted to $S$ and smaller (i.e. coarser) than or equal to $\mathscr{C}$. Moreover, when $n=1$ or $n=2$, we strengthen this result by proving the existence of a minimum element in CAD$(S)$. Astonishingly for $n \geq 3$, there exist semi-algebraic sets whose associated poset of adapted CADs does not admit a minimum. We prove this result by providing explicit examples. We finally use a reduction relation on CAD$(S)$ to define an algorithm for the computation of minimal CADs. We conclude with a characterization of those semi-algebraic sets $S$ for which CAD$(S)$ has a minimum by means of confluence of the associated reduction system.

cs.SC↗