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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 \(Z (\sum_{a \in A} c_a t^{\omega (a)} x^a)\), where \(A \subseteq \mathbb{Z}^n\) is finite and \(\omega : A \to \mathbb{Z}\) a function defining a triangulation of \(\text{conv} (A)\), 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 \(\mathbb{P}^3 (\mathbb{R})\).

MSC:
14M10 Complete intersections
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14P25 Topology of real algebraic varieties
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
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References:
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