An arithmetic enrichment of Bézout’s theorem. (English) Zbl 1467.14126
Summary: 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 J. L. Kass and K. Wickelgren [Compos. Math. 157, No. 4, 677–709 (2021; Zbl 1477.14085)], 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.
MSC:
| 14N15 | Classical problems, Schubert calculus |
| 14F42 | Motivic cohomology; motivic homotopy theory |
| 11E81 | Algebraic theory of quadratic forms; Witt groups and rings |
Citations:
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