Cylindrical Algebraic Decomposition and connected components
From what I've read Sage has support for accessing QEPCAD which implements Cylindrical Algebraic Decomposition.
Given a collection of polynomial inequalities I would like to know if they define a single connected region or multiple regions. It looks to me that the Cylindrical Algebraic Decomposition can be used to determine that though I don't know how to get at that. Does QEPCAD support that question or do I need to understand a lot of details to implement that using QEPCAD?
From reading documentation at https://doc.sagemath.org/html/en/refe... the answer appears to be Yes. Just count how many cells have required polynomial signs and check if this number is 1.
I don't think cells represent largest connected regions.
Why? When we cross a boundary between two cells, the inequality corresponding to this boundary changes its sign. Hence, if there is a path connecting two points and preserving all inequalities along the way, it should not cross any cell boundaries, meaning that such a path must be entirely within a single cell.