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.

中文翻译：

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Bézout 定理的算术扩充

Bézout 定理的经典版本给出了代数闭域上射影空间中超曲面的交点的整数值计数。使用 Kass 和 Wickelgren 的工作，我们通过给出射影空间中超曲面的交点的双线性形式值计数，在任何完美域上证明了 Bézout 定理的一个版本。在非代数闭域上，这个丰富的 Bézout 定理在超曲面的交点处的梯度上强加了一个关系。作为推论，我们在奇数特征的实数、有理数和有限域上获得了 Bézout 定理的算术-几何版本。