Graphs and Combinatorics

, Volume 27, Issue 3, pp 419–430

A Note on the Complexity of Real Algebraic Hypersurfaces

Proceedings Paper

DOI: 10.1007/s00373-011-1020-7

Cite this article as:
Kerber, M. & Sagraloff, M. Graphs and Combinatorics (2011) 27: 419. doi:10.1007/s00373-011-1020-7


Given an algebraic hypersurface \({\fancyscript{O}}\) in \({\mathbb{R}^d}\), how many simplices are necessary for a simplicial complex isotopic to \({\fancyscript{O}}\)? We address this problem and the variant where all vertices of the complex must lie on \({\fancyscript{O}}\). We give asymptotically tight worst-case bounds for algebraic plane curves. Our results gradually improve known bounds in higher dimensions; however, the question for tight bounds remains unsolved for d ≥ 3.


Algebraic curves Algebraic surfaces Triangulation Isotopy 

Copyright information

© Springer 2011

Authors and Affiliations

  1. 1.IST (Institute of Science and Technology) AustriaKlosterneuburgAustria
  2. 2.Max-Planck-Institute for InformaticsSaarbrückenGermany

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