论文标题
均质多项式的F-Pure阈值的值
Values of the F-pure threshold for homogeneous polynomials
论文作者
论文摘要
我们在n变量的特征p的代数闭合字段上，在n变量中的f-pure阈值的值中找到了一个公式，用于f-pure阈值的值。我们还表明，在每个特征p和所有d（大于3）的特征性p中，总有降低的D度d的多项式存在，其f-pure阈值是在某个地方的基本p扩展为2/d的截断。特别是，始终存在降低的多项式，其f-pure阈值严格小于2/d。我们提供了一个示例来消极地解决Hernandez，Núñez-Betancourt，Witt和Zhang提出的问题，即它们是否证明了它们对降低形式的F-Pure阈值的列表是“最小p”。另一方面，我们还提供了支持和完善他们想法的证据，包括确定基本p扩展为2/d的特定截断，这些截断始终是降低的D学位形式的F-Pure阈值，以及显示其条件（以每个特征）的计算，最多八个和其他几个情况。最后，我们指出了在其程度和特征p的降低形式的F-Pure阈值上的下限。
We find a formula, in terms of n, d and p, for the value of the F-pure threshold for the generic homogeneous polynomial of degree d in n variables over an algebraically closed field of characteristic p. We also show that, in every characteristic p and for all d (greater than 3) not divisible by p, there always exist reduced polynomials of degree d whose F-pure threshold is a truncation of the base p expansion of 2/d at some place; in particular, there always exist reduced polynomials whose F-pure threshold is strictly less than 2/d. We provide an example to resolve, negatively, a question proposed by Hernandez, Núñez-Betancourt, Witt and Zhang, as to whether a list of necessary restrictions they prove on the F-pure threshold of reduced forms are "minimal" for large p. On the other hand, we also provide evidence supporting and refining their ideas, including identifying specific truncations of the base p expansion of 2/d that are always F-pure thresholds for reduced forms of degree d, and computations that show their conditions suffice (in every characteristic) for degrees up to eight and several other situations. Finally, we point out a lower bound on the F-pure threshold of a reduced form in terms of its degree and the characteristic p.
