On real algebraic realization of round fold maps of codimension $-1$
Naoki Kitazawa
TL;DR
This work addresses the problem of real algebraic realization for round fold maps of codimension $-1$, bridging singularity theory with real algebraic geometry. It develops a framework built on Reeb graphs, Morse/M-digraphs, and Poincaré-Reeb graphs, and provides explicit polynomial constructions so that the zero set $M$ inside ${\mathbb R}^{m+1}$ yields a round fold map via ${\pi}_{m+1,m-1}|_M$. The contributions include refined Main Theorems that realize round fold maps as real algebraic objects using products of polynomials $h_i$ (and $H_S$ in certain cases) and characterize the associated algebraic domains, SM-digraphs, and trivial spinning constructions. The methods advance real algebraic geometry in tandem with singularity theory, offering concrete algebraic models for round fold maps and clarifying how topological data of the domain manifold are encoded in polynomial data.
Abstract
The canonical projections of the unit spheres are generalized to special generic maps and round fold maps, for example. They are generalizations from the viewpoint of singularity theory of differentiable maps and these maps restrict the topologies and the differentiable structures of the manifolds. We are concerned with round fold maps, defined as smooth maps locally represented as the product map of a Morse function and the identity map on a smooth manifold, and maps with singular value sets being concentric spheres. A bit different from differential topology, we are concerned with real algebraic geometric aspects of these maps. We discuss real algebraic realization of round fold maps of codimension $-1$ as our new work. Real algebraic realization of these maps is of fundamental and important studies in real algebraic geometry and a new study recently developing mainly due to the author.