An arithmetic enrichment of B\'ezout's Theorem

The classical version of B'ezout’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 B'ezout’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 B'ezout’s Theorem imposes a relation on the gradients of the hypersurfaces at their intersection points. As corollaries, we obtain arithmetic-geometric versions of B'ezout’s Theorem over the reals, rationals, and finite fields of odd characteristic.