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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Closed planar curves without inflections
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by Shuntaro Ohno, Tetsuya Ozawa and Masaaki Umehara PDF
Proc. Amer. Math. Soc. 141 (2013), 651-665 Request permission


We define a computable topological invariant $\mu (\gamma )$ for generic closed planar regular curves $\gamma$, which gives an effective lower bound for the number of inflection points on a given generic closed planar curve. Using it, we classify the topological types of locally convex curves (i.e. closed planar regular curves without inflections) whose numbers of crossings are less than or equal to five. Moreover, we discuss the relationship between the number of double tangents and the invariant $\mu (\gamma )$ of a given $\gamma$.
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Additional Information
  • Shuntaro Ohno
  • Affiliation: Department of Mathematics, Kyoto Koka Senior High School, Nishi-kyogoku Nodacho, Kyoto 615-0861, Japan
  • Email:
  • Tetsuya Ozawa
  • Affiliation: Department of Mathematics, Meijo University, Tempaku, Nagoya, 468-8502 Japan
  • Email:
  • Masaaki Umehara
  • Affiliation: Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1-W8-34, O-okayama Meguro-ku, Tokyo 152-8552, Japan
  • MR Author ID: 237419
  • Email:
  • Received by editor(s): December 21, 2010
  • Received by editor(s) in revised form: June 21, 2011, and June 26, 2011
  • Published electronically: June 1, 2012
  • Additional Notes: The third author was partially supported by the Grant-in-Aid for Scientific Research (A) No. 22244006, Japan Society for the Promotion of Science.
  • Communicated by: Daniel Ruberman
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 141 (2013), 651-665
  • MSC (2010): Primary 53A04; Secondary 53A15, 53C42
  • DOI:
  • MathSciNet review: 2996970