Reconstruction of real algebraic functions from finite V-digraphs

Reconstruct nice and explicit real algebraic functions whose Reeb V-digraphs are isomorphic to a given finite V-digraph.

Background

The paper associates arrangements of circles with Poincaré–Reeb V-digraphs and realizes the closures of the associated planar regions as images of natural real algebraic maps. This motivates the inverse problem of starting with a prescribed finite V-digraph and constructing an explicit real algebraic function that has the prescribed Reeb-graph structure.

The problem is presented as a continuation of earlier work and is identified as having originated in the differentiable category, with related results cited for smooth functions. The unresolved issue is whether every given finite V-digraph, or which such digraphs, can be realized by explicit real algebraic functions.