Real Algebraic and Analytic Geometry

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237. Benoit Bertrand and Frédéric Bihan:
Euler characteristic of real non degenerate tropical complete intersections.

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Submission: 2007, November 6.

We define nondegenerate tropical complete intersections imitating the corresponding definition in complex algebraic geometry. As in the complex situation, all nonzero intersection multiplicity numbers between tropical hypersurfaces defining a nondegenerate tropical complete intersection are equal to $1$. The intersection multiplicity numbers we use are sums of mixed volumes of polytopes which are dual to cells of the tropical hypersurfaces. We show that the Euler characteristic of a real nondegenerate tropical complete intersection depends only on the Newton polytopes of the tropical polynomials which define the intersection. Basically, it is equal to the usual signature of a complex complete intersection with same Newton polytopes, when this signature is defined. The proof reduces to the toric hypersurface case, and uses the notion of $E$-polynomials of complex varieties.

Full text, 30p.: pdf 324k.

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