2025/07/13 by Naoki Kitazawa, Kitazawa, Naoki
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Metric Geometry (math.MG) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2507.09467
openalex publication_date 2025/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We discuss reconstructing smooth real algebraic maps onto curves whose Reeb graph is as prescribed. %This can be contribut\ed 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.