Arrangements of circles supported by small chords and compatible with natural real algebraic functions

TL;DR

This paper explores new arrangements of small circles compatible with natural real algebraic functions, focusing on their topological properties, combinatorics, and applications in constructing algebraic maps and manifolds.

Contribution

It introduces a novel class of circle arrangements related to real algebraic maps, extending previous work and providing explicit constructions and topological insights.

Findings

Complete classification of local changes in arrangements

Construction of real algebraic maps with prescribed arrangements

Analysis of topological and combinatorial properties

Abstract

We have previously proposed a study of arrangements of small circles which also surround regions in the plane realized as the images of natural real algebraic maps yielding Morse-Bott functions by projections. Among studies of arrangements, families of smooth regular submanifolds in smooth manifolds, this study is fundamental, explicit, and new, surprisingly. We have obtained a complete list of local changes of the graphs the regions naturally collapse to in adding a (generic) small circle to an existing arrangement of the proposed class. Here, we propose a similar and essentially different class of arrangements of circles. The present study also yields real algebraic maps and nice real algebraic functions similarly and we present a similar study. We are interested in topological properties and combinatorics among such arrangements and regions and applications to constructing such real algebraic maps and manifolds explicitly and understanding their global structures.