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Poincar\'e-Reeb graphs of real algebraic domains

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arxiv 2207.06871 v2 pith:RBKXZRI6 submitted 2022-07-14 math.GT

Arnaud Bodin , Patrick Popescu-Pampu , Miruna-Stefana Sorea This is my paper · ORCID
classification math.GT
keywords algebraice-reebgraphpoincarrealverticaldomainlines
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An algebraic domain is a closed topological subsurface of a real affine plane whose boundary consists of disjoint smooth connected components of real algebraic plane curves. We study the geometric shape of an algebraic domain by collapsing all vertical segments contained in it: this yields a Poincar\'e-Reeb graph, which is naturally transversal to the foliation by vertical lines. We show that any transversal graph whose vertices have only valencies 1 and 3 and are situated on distinct vertical lines can be realized as a Poincar\'e-Reeb graph.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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

    math.AG 2025-01 conditional novelty 6.0 of 10

    SSC-NI arrangements are defined, and a complete classification of local Poincaré-Reeb V-digraph changes under chord-supported circle additions is asserted, with an example not realizable by the previous MBCC class.

  2. Arrangements of small circles for Morse-Bott functions

    math.AG 2024-12 conditional novelty 6.0 of 10

    The paper classifies the local changes to Poincaré-Reeb graphs caused by adding small circles centered on existing circles, for circle arrangements associated with Morse-Bott functions.

  3. Graphs with tree decompositions of small graphs and realizing them as the Reeb graphs of real algebraic functions

    math.AG 2025-08 conditional novelty 4.0 of 10

    Every tree, and certain graphs assembled from single edges and small circles, is the Reeb graph of a Morse-Bott real algebraic function defined by degree-1 and degree-2 polynomials.

  4. Reconstruction of real algebraic functions into curves with prescribed Reeb graphs

    math.AG 2025-07 conditional novelty 4.0 of 10

    For any finite graph satisfying genericity conditions and any dimension at least 2, the paper constructs real algebraic maps to curves whose Reeb graph is isomorphic to the graph.

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