  Assume D is a simplicial sphere, and k_1 is a field. We say that D is
generically anisotropic over k_1 if, for a certain purely transcendental field
extension k of k_1, a certain Artinian reduction A of the Stanley-Reisner ring
k[D] has the following property: All nonzero homogeneous elements u of A of
degree less or equal to (dim D +1)/2 have nonzero square. We prove, using
suitable differential operators, that, if the field k_1 has characteristic 2,
then every simplicial sphere D is generically anisotropic over k_1. As an
application, we give a second proof of a recent result of Adiprasito, known as
McMullen's g-conjecture for simplicial spheres. We also prove that the
simplicial spheres of dimension 1 are generically anisotropic over any field
k_1.
