论文标题
Bézout定理的算术丰富
An arithmetic enrichment of Bézout's Theorem
论文作者
论文摘要
Bézout定理的经典版本给出了在代数封闭的字段上投影空间中高空曲面相交点的整数值计数。使用Kass和Wickelgren的作品，我们通过在投射空间中的Hypersurfaces的交叉点进行双线性表单值来证明Bézout定理的版本。在非广场封闭的磁场上，这种丰富的Bézout定理在其交叉点上对Hypersurfaces的梯度施加了关系。作为推论，我们获得了Bézout定理的算术几何版本，这些版本是奇数特征的真实，理性和有限领域。
The classical version of Bézout's Theorem gives an integer-valued count of the intersection points of hypersurfaces in projective space over an algebraically closed field. Using work of Kass and Wickelgren, we prove a version of Bézout's Theorem over any perfect field by giving a bilinear form-valued count of the intersection points of hypersurfaces in projective space. Over non-algebraically closed fields, this enriched Bézout's Theorem imposes a relation on the gradients of the hypersurfaces at their intersection points. As corollaries, we obtain arithmetic-geometric versions of Bézout's Theorem over the reals, rationals, and finite fields of odd characteristic.
