00:00:00 / 00:00:00

Appears in collection : Real Algebraic Geometry / Géometrie algébrique réelle

A real algebraic domain is a closed topological subsurface of a real affine plane such that its boundary consists of disjoint smooth connected components of real algebraic plane curves. Our goal is to study the nonconvexity of real algebraic domains relative to the vertical direction. To this end, we collapse all vertical segments contained in the algebraic domain, yielding a Poincar´e–Reeb graph which is naturally transversal to the foliation by vertical lines. Our main result is the following: any transversal graph whose vertices have only valencies 1 and 3 and are situated on distinct vertical lines arises up to isomorphism as a Poincar´e–Reeb graph of a real algebraic domain. We also give a purely topological description of the setting in which our construction of Poincar´e–Reeb graphs may be applied, with no differentiability assumptions. This is a joint work with Arnaud Bodin and Patrick Popescu-Pampu (Université de Lille, France).

Information about the video

Citation data

  • DOI 10.24350/CIRM.V.19978103
  • Cite this video Sorea, Miruna-Stefana (25/10/2022). Poincaré-Reeb graphs of real algebraic domains. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19978103
  • URL https://dx.doi.org/10.24350/CIRM.V.19978103

Domain(s)

Bibliography

  • Saugata Basu, Richard Pollack, and Marie-Françoise Roy, Algorithms in real algebraic geometry, second ed., Algorithms and Computation in Mathematics, vol. 10, Springer-Verlag, Berlin, 2006. - http://dx.doi.org/10.1007/3-540-33099-2
  • Riccardo Benedetti and Jean-Jacques Risler, Real algebraic and semi-algebraic sets, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1990.
  • Nate Elredge, Answer to On finding polynomials that approximate a function and its derivative, StackExchange, question 555712 (2013). - https://math.stackexchange.com/questions/555712/on-finding-polynomials-that-approximate-a-function-and-its-derivative-extension
  • Étienne Ghys, A singular mathematical promenade, ENS Éditions, Lyon, 2017.
  • Henri Poincaré, Papers on topology, cinquième complément à l'analysis situs, History of Mathematics, vol. 37, American Mathematical Society, Providence, RI; London Mathematical Society, London, 2010, Rendiconti del Circolo Matematico di Palermo (1884-1940), 45–110, Springer, translated and with an introduction by John Stillwell, 1904.
  • Georges Reeb, Sur les points singuliers d'une forme de Pfaff complètement intégrable ou d'une fonction numérique, C. R. Acad. Sci. Paris 222 (1946), 847–849.
  • Miruna-Ştefana Sorea, The shapes of level curves of real polynomials near strict local minima, Ph.D. thesis, Université de Lille/Laboratoire Paul Painlevé, 2018. - https://tel.archives-ouvertes.fr/tel-01909028
  • Miruna-Ştefana Sorea, Constructing separable Arnold snakes of Morse polynomials, Port. Math. 77 (2020), no. 2, 219–260. - https://doi.org/10.4171/pm/2050
  • Miruna-Ştefana Sorea, Measuring the local non-convexity of real algebraic curves, J. Symbolic Comput. 109 (2022), 482–509 - https://doi.org/10.1016/j.jsc.2020.07.017
  • Miruna-Ştefana Sorea, Permutations encoding the local shape of level curves of real polynomials via generic projections, Annales de l'Institut Fourier, Volume 72 (2022) no. 4, pp. 1661-1703. - https://doi.org/10.5802/aif.3479
  • Marshall Harvey Stone, The generalized Weierstrass approximation theorem, Math. Mag. 21 (1948), 167–184, 237–254. - http://dx.doi.org/10.2307/3029337
  • Oleg Yanovich Viro, Some integral calculus based on Euler characteristic, Topology and geometry—Rohlin Seminar, Lecture Notes in Math., vol. 1346, Springer, Berlin, 1988, pp. 127–138 - http://dx.doi.org/10.1007/BFb0082775
  • Charles Terence Clegg Wall, Singular points of plane curves, London Mathematical Society Student Texts, vol. 63, Cambridge University Press, Cambridge, 2004 -

Last related questions on MathOverflow

You have to connect your Carmin.tv account with mathoverflow to add question

Ask a question on MathOverflow




Register

  • Bookmark videos
  • Add videos to see later &
    keep your browsing history
  • Comment with the scientific
    community
  • Get notification updates
    for your favorite subjects
Give feedback