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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 $A \subseteq \bbfZ^n$ is finite and $\omega : A \to \bbfZ$ a function defining a triangulation of $\text{conv} (A)$, is homeomorphic to a certain simplicial complex for small $t > 0$ [cf. {\it 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 {\it 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 $\bbfP^3 (\bbfR)$.

##### MSC:
 14M10 Complete intersections 14M25 Toric varieties, Newton polyhedra 14P25 Topology of real algebraic varieties 14N10 Enumerative problems (algebraic geometry)
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##### References:
 [1] I.M. Gel’fand - M.M. Kapranov - A.V. Zelevinsky , Discriminants, Resultants, and Multidimensional Determinants . Birkhäuser , Bòston , 1994 . MR 1264417 | Zbl 0827.14036 · Zbl 0827.14036 [2] I.M. Gel’fand - M.M. Kapranov - A.V. Zelevinsky , Discriminants of polynomials in several variables and triangulations of Newton polytopes. Algebra i analiz ( Leningrad Mathematical Journal ) 2 ( 1990 ), 1 - 62 . MR 1052262 | Zbl 0741.14033 · Zbl 0741.14033 [3] C. Lee , Regular triangulations of convex polytopes . In: Applied Geometry and Discrete Mathematics The Victor Klee Festschrift , P. Gritzmann - B. Sturmfels eds., American Math. Soc ., DIMACS Series 4 , Providence, R.I ., 1991 , 443 - 456 . MR 1116369 | Zbl 0746.52015 · Zbl 0746.52015 [4] J.-J. Risler , Construction d’hypersurfaces reélles ayant une topologie donnée [d’aprés Viro], Séminaire N. Bourbaki, exposé no. 762 , Masson , Paris , 1992 - 93 . [5] B. Sturmfels , The asymptotic number of real zeros of a sparse polynomial system. To appear in: Proceedings of the Workshop on ”Hamiltonian and Gradient Flows: Algorithms and Control” , Fields Institute , Waterloo, Ontario , March 1992 . [6] B. Sturmfels , On the Newton polytope of the resultant , Journal of Algebraic Combinatorics 3 ( 1994 ), 207 - 236 . MR 1268576 | Zbl 0798.05074 · Zbl 0798.05074 · doi:10.1023/A:1022497624378 [7] O.Ya. Viro , Gluing of algebraic hypersurfaces, removing of singularities and constructions of curves . In: Proceedings of the International Topology Conference at Leningrad , 1983 (in Russian), 149 - 197 . Zbl 0605.14021 · Zbl 0605.14021 [8] O.Ya. Viro , Gluing of plane real algebraic curves and construction of curves of degrees 6 and 7. In ”Topology” , (L.D. Faddeev - A.A. Mal’cev eds.), Springer Lectures Notes in Mathematics 1060 , 1984 , pp. 187 - 200 . MR 770238 | Zbl 0576.14031 · Zbl 0576.14031 [9] O.Ya. Viro , Real algebraic plane curves: constructions with controlled topology , Leningrad Mathematical Journal 1 ( 1990 ), 1059 - 1134 . MR 1036837 | Zbl 0732.14026 · Zbl 0732.14026 [10] V.I. Danilov - A.G. Khovanskii , Newton polyhedra and an algorithm for computing Hodge-Deligne numbers , Math. USSR-Izv. 29 ( 1987 ), 279 - 298 . Zbl 0669.14012 · Zbl 0669.14012 · doi:10.1070/IM1987v029n02ABEH000970