Realization of graphs by real algebraic functions

Reconstruct, for a given graph, a real algebraic function on some connected component of the zero set of a polynomial map.

Background

The paper recalls that the Reeb graph of a smooth function encodes the connected components of its level sets and has applications to compactly representing manifolds. It then relates this classical construction to the Poincaré–Reeb graphs studied for regions surrounded by circles.

The unresolved question asks whether an arbitrary graph can be obtained from a real algebraic function defined on a connected component of the zero set of a polynomial map, extending the paper’s motivation from smooth and differential-topological settings to real algebraic geometry.