单纯球的特征2各向异性
The characteristic 2 anisotropicity of simplicial spheres
摘要：
设D是一个单形球，k_1是一个域。我们说D是k_1上的一般各向异性的，如果对于k_1的某个纯超越域扩张k，Stanley-Reisner环k[D]的某个Artin约化A具有以下性质：A的次小于或等于（dim D +1）/2的所有非零齐次元素u有非零平方。利用适当的微分算子证明了：若域k_1具有特征2，则每个单形球面D在k_1上是泛各向异性的.作为一个应用程序，我们给出了第二个证明最近的结果Adiprasito，被称为McMullen的g-猜想的单纯球。我们还证明了1维单纯球在任意域k_1上是泛各向异性的。
英文摘要：
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.
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