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Assume D is a simplicial sphere, and k_1 is a field. We say that D isgenerically anisotropic over k_1 if, for a certain purely transcendental fieldextension k of k_1, a certain Artinian reduction A of the Stanley-Reisner ringk[D] has the following property: All nonzero homogeneous elements u of A ofdegree less or equal to (dim D +1)/2 have nonzero square. We prove, usingsuitable differential operators, that, if the field k_1 has characteristic 2,then every simplicial sphere D is generically anisotropic over k_1. As anapplication, we give a second proof of a recent result of Adiprasito, known asMcMullen's g-conjecture for simplicial spheres. We also prove that thesimplicial spheres of dimension 1 are generically anisotropic over any fieldk_1.

The characteristic 2 anisotropicity of simplicial spheres