Reconstruction of real algebraic functions into curves with prescribed Reeb graphs

TL;DR

This paper presents a method for reconstructing smooth real algebraic maps onto curves with prescribed Reeb graphs, extending previous work to the curve-valued case and contributing new techniques to real algebraic geometry.

Contribution

It introduces a novel approach for reconstructing real algebraic functions into curves with specific Reeb graphs, especially focusing on the curve-valued case.

Findings

Reconstruction of functions from finite graphs is achieved.

Extension of Reeb graph theory to real algebraic maps onto curves.

Provides explicit examples in real algebraic geometry.

Abstract

We discuss reconstructing smooth real algebraic maps onto curves whose Reeb graph is as prescribed. This can be contributed to real algebraic geometry, especially in explicit examples in real algebraic geometry in a new way. The Reeb graph of a smooth function is the space of all connected components of preimages of all single points and a natural quotient space of the manifold with the vertex set being all connected components containing some singular points of it. This gives a strong tool in geometry of manifolds and appeared already in 1950 with Morse functions. The Reeb graph of the natural height of the unit sphere of dimension at least 2 is a graph with exactly two vertices and one edge. We reconstruct functions, from general finite graphs, conversely. In the differentiable situations, Sharko pioneered this in 2006, followed by Masumoto-Saeki and Michalak, mainly. Related real algebraic situations have been launched and studied by the author. The curve-valued case is first considered here.