Geodesic
The Euler Characteristic of a Complete Intersection in Terms of the Newton Polyhedra Revisited
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Toric Topology, Group Actions, Geometry, and Combinatorics. Part 2, Tome 318 (2022), pp. 66-72

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The well-known formula for the Euler characteristic of a complete intersection in the complex torus in terms of the supports of the Laurent polynomials, the left-hand sides of the defining equations (in fact, in terms of the convex hulls of these supports, Newton polyhedra), was announced in a short note by D. N. Bernshtein, A. G. Kushnirenko, and A. G. Khovanskii (1976). The proof of the formula was given by A. G. Khovanskii (1978), but it was not self-contained (it was based on results of another author) and was somewhat fragmentary. Here we give a more elementary proof of this equation based on the simplest properties of toric manifolds.
Keywords: Newton polyhedron, complete intersection, Euler characteristic, toric manifold. 
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S. M. Gusein-Zade. The Euler Characteristic of a Complete Intersection in Terms of the Newton Polyhedra Revisited. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Toric Topology, Group Actions, Geometry, and Combinatorics. Part 2, Tome 318 (2022), pp. 66-72. http://geodesic.mathdoc.fr/item/TM_2022_318_a5/