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Geometric and Functional Analysis

, Volume 27, Issue 1, pp 33–66 | Cite as

Signs of the Leading Coefficients of the Resultant

  • Arina Arkhipova
  • Alexander Esterov
Article

Abstract

We construct a certain \({\mathbb{F}_{2}}\)-valued analogue of the mixed volume of lattice polytopes. This 2-mixed volume cannot be defined as a polarization of any kind of an additive measure, or characterized by any kind of its monotonicity properties, because neither of the two makes sense over \({\mathbb{F}_2}\). In this sense, the convex-geometric nature of the 2-mixed volume remains unclear. On the other hand, the 2-mixed volume seems to be no less natural and useful than the classical mixed volume—in particular, it also plays an important role in algebraic geometry. As an illustration of this role, we obtain a closed-form expression in terms of the 2-mixed volume to compute the signs of the leading coefficients of the resultant, which were by now explicitly computed only for some special cases.

Keywords and phrases

Convex geometry Algebraic geometry Tropical geometry Mixed volumes Newton polyhedra Resultants 

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Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.National Research University Higher School of EconomicsMoscowRussia

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