Signs of the Leading Coefficients of the Resultant

The polarization construction establishes a one-to-one correspondence between polynomial and multilinear forms over fields of characteristic 0. A special case of importance is to construct the mixed versions for various measures of convex bodies. For instance, the mixed volume, which plays a significant role in algebraic geometry, is the polarization of the volume with respect to the Minkowski addition. We construct an analogue of the mixed volume of lattice polytopes, which takes values in $\F_2$. This 2-mixed volume is remarkable, since it cannot be obtained by means of the standard polarization construction, and, on the other hand, is no less essential 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.