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arXiv · 2003.07413

An arithmetic enrichment of Bézout's Theorem

Stephen McKean
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Abstract

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.

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Stephen McKean. 2020-11-04. An arithmetic enrichment of Bézout's Theorem. https://doi.org/10.1007/s00208-020-02120-3

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