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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2022, Volume 318, Pages 66–72
DOI: https://doi.org/10.4213/tm4261 (Mi tm4261) 		 
The Euler Characteristic of a Complete Intersection in Terms of the Newton Polyhedra Revisited

S. M. Gusein-Zadeabc

a Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, 119991 Russia
b Moscow Center of Fundamental and Applied Mathematics, Moscow, 119991 Russia
c Faculty of Mathematics, HSE University, ul. Usacheva 6, Moscow, 119048 Russia
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DOI: https://doi.org/10.4213/tm4261
Abstract: 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.
Funding agency Grant number
Russian Science Foundation 21-11-00080
This work was supported by the Russian Science Foundation under grant no. 21-11-00080 and performed at Lomonosov Moscow State University.
Received: February 26, 2022
Revised: April 12, 2022
Accepted: April 22, 2022
English version:
Proceedings of the Steklov Institute of Mathematics, 2022, Volume 318, Pages 59–64
DOI: https://doi.org/10.1134/S008154382204006X
Bibliographic databases:
Document Type: Article
UDC: 514.76+515.165
Language: Russian
Citation: S. M. Gusein-Zade, “The Euler Characteristic of a Complete Intersection in Terms of the Newton Polyhedra Revisited”, Toric Topology, Group Actions, Geometry, and Combinatorics. Part 2, Collected papers, Trudy Mat. Inst. Steklova, 318, Steklov Math. Inst., Moscow, 2022, 66–72; Proc. Steklov Inst. Math., 318 (2022), 59–64
Citation in format AMSBIB
\Bibitem{Gus22}
\by S.~M.~Gusein-Zade
\paper The Euler Characteristic of a Complete Intersection in Terms of the Newton Polyhedra Revisited
\inbook Toric Topology, Group Actions, Geometry, and Combinatorics. Part~2
\bookinfo Collected papers
\serial Trudy Mat. Inst. Steklova
\yr 2022
\vol 318
\pages 66--72
\publ Steklov Math. Inst.
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm4261}
\crossref{https://doi.org/10.4213/tm4261}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2022
\vol 318
\pages 59--64
\crossref{https://doi.org/10.1134/S008154382204006X}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85142011139}
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    		Труды Математического института имени В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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