Signs of the Leading Coefficients of the Resultant

Abstract: We construct a certain $\F_2$-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 $\F_2$. 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.