A Note on the Complexity of Real Algebraic Hypersurfaces

Abstract

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

CW Complexes for Complex Algebraic Surfaces

“…See also [1], and especially the introduction, with its thorough overview and references. As mentioned there, an analysis of cylindrical algebraic decomposition [5], an algorithm that has been both well studied and widely implemented, yields a bound of O(d 2 n +3 n ) for a degree-d real algebraic hypersurface of dimension n. A recent preprint [13] focuses on the case of hypersurfaces and delivers triangulations with O(d 3·2 n−1 −1 ) cells.…”