Abstract
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.
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Kerber, M., Sagraloff, M. A Note on the Complexity of Real Algebraic Hypersurfaces. Graphs and Combinatorics 27, 419–430 (2011). https://doi.org/10.1007/s00373-011-1020-7
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DOI: https://doi.org/10.1007/s00373-011-1020-7