Viro theorem and topology of real and complex combinatorial hypersurfaces

Geometric bistellar flips: the setting, the context and a construction

“…Regular triangulations are particularly simple and yet quite versatile. They appear in different contexts under different names such as coherent [29], convex [36], [77], Gale [49], or generalized (or, weighted ) Delaunay [25] triangulations. The latter refers to the fact that the Delaunay triangulation of A, probably the most used triangulation in applications, is the regular triangulation obtained with w(a) = a 2 , where · is the euclidean norm.…”

A Viro theorem without convexity hypothesis for trigonal curves

“…Here, we explain the patchworking method for curves. The general patchworking Theorem, which uses the convexity of the subdivision, can be found in [Vir84], [Vir89], [Vir], [Ris92] and [IS03]. Our patchwork Theorem, Theorem 3.4, does not require this convexity assumption, yet it works only for trigonal curves.…”

“…Step 1. In the patchwork construction, one can use complexification of lattice polygons and complex equivariant charts of real polynomials as defined in [IS02], [IS03] and [Vir]. Then, the patchwork procedure gives a piecewise smooth surface, invariant under the action of the complex conjugation, in the complexification of the triangle with vertices (0, 0), (0, 3) and (3n, 0).…”

The Viro Method for Construction of Piecewise Algebraic Hypersurfaces

“…Central to these developments is a combinatorial construction due to the Viro method [4][5][6]. The Viro method is a powerful construction method of real nonsingular algebraic hypersurfaces with prescribed topology (see [4][5][6][7][8][9][10][11][12][13][14]). It provides a link between the topology of real algebraic varieties and toric varieties.…”