Refined algebraic domains with finite sets in the boundaries respecting differential geometry

TL;DR

The paper addresses realizing refined algebraic domains—regions bounded by real algebraic curves with characteristic finite sets derived from differential geometry, including inflection points and double tangent lines—within the plane. It develops Morse-like refinements and Poincaré-Reeb graphs associated to projection maps such as ${\pi}_{2,1}$ (and ${\pi}_{2,1,2}$), and proves existence theorems (Main Theorems 1 and 2) showing that Morse refined algebraic domains can be constructed to realize prescribed graphs up to isomorphism, via perturbations near critical boundary features and by approximating smooth curves with real algebraic ones while preserving graph structure. The results extend previous work by Kitazawa on realizing graphs from algebraic curves and provide explicit algebraic models for boundary-graph correspondences beyond simple shapes like lines, circles, or ellipsoidal boundaries. Together these findings connect real algebraic geometry, differential geometry of curves, and combinatorial graph theory to enable controlled, explicit constructions of boundary configurations with desired Poincaré-Reeb graphs.

Abstract

We are interested in shapes of real algebraic curves in the plane and regions surrounded by them: they are named refined algebraic domains by the author. As characteristic finite sets, we consider points contained in two curves and the sets of singular points of the restrictions of the projections to the lines to the curves. As a new case, we respect differential geometry and consider inflection points and points of some double tangent lines of a single connected curve. We prove fundamental properties and investigate some examples. We have also previously considered the cases where the curves are straight lines, circles, or boundaries of ellipsoids for example. Such simple cases are trivial in our new consideration.

Figures (2)

• Figure 1: The region $\overline{D_{\mathcal{S}}}$ is colored in black. The blue colored point shows an inflection point $p$ of a smooth connected curve $S_j$ in the first case. The blue colored point shows a point in a double tangent line of the smooth connected curve $S_j$ in the second case. These points $p$ are also points of $\overline{D_{\mathcal{S}}}-D_{\mathcal{S}}$. Blue colored dotted segments show tangent lines to the points $p$. The two points $p_{\rm l} \neq p$ and $p_{\rm r} \neq p$ are colored in green. In the first case, they are connected by a smooth curve, colored in green, and sufficiently close to $p$.
• Figure 2: The regions $\overline{D_{\mathcal{S}}}$ are colored in gray. The blue colored points show the points of $p \in F_{D_{\mathcal{S}},0}$ which may be inflection points of smooth connected curves $S_j$ or points in some double tangent lines of the smooth connected curves $S_j$. The two points $p_{\rm l} \neq p$ and $p_{\rm r} \neq p$ are colored in green and connected by straight segments, dotted in blue and sufficiently close to $p$.

Theorems & Definitions (3)

• proof : A proof of Main Theorem \ref{['mthm:1']}
• proof
• proof