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Viro’s theorem for complete intersections. (English) Zbl 0826.14032
Viro’s theorem for hypersurfaces says that the real algebraic set Z( aA c a t ω(a) x a ), where A n is finite and ω:A a function defining a triangulation of conv(A), is homeomorphic to a certain simplicial complex for small t>0 [cf. O. Ya. Viro in: Topology Conf., Proc., Collect. Rep., Leningrad 1982, 149-197 (1983; Zbl 0605.14021)]. The author generalizes this theorem to arbitrary complete intersections using his theory of mixed decompositions of Newton polytopes introduced by B. Sturmfels [J. Algebr. Comb. 3, No. 2, 207-236 (1994; Zbl 0798.05074)]. Applications include the study of the number of real points of zero-dimensional complete intersections, and of the topology of complete intersection curves in 3 ().

14M10Complete intersections
14M25Toric varieties, Newton polyhedra
14P25Topology of real algebraic varieties
14N10Enumerative problems (algebraic geometry)