The Euler Characteristic of a Complete Intersection in Terms of the Newton Polyhedra Revisited

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...

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Bibliographic Details
Published in:Proceedings of the Steklov Institute of Mathematics 2022-09, Vol.318 (1), p.59-64
Main Author: Gusein-Zade, S. M.
Format: Article
Language:English
Subjects:
14/34
639/766/189
639/766/530
639/766/747
Convexity
Formulas (mathematics)
Intersections
Mathematics
Mathematics and Statistics
Polyhedra
Polynomials
Toruses
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Summary: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.
ISSN:0081-5438
1531-8605
DOI:10.1134/S008154382204006X