An arithmetic enrichment of Bézout's Theorem

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.

arXiv commentUpdated and revised version for publication. 27 pages, 3 figures

References in corpus (2)

• The class of Eisenbud--Khimshiashvili--Levine is the local A1-Brouwer degree
• Motivic Euler characteristics and Witt-valued characteristic classes

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• Conics meeting eight lines over perfect fields
• Ulrich sheaves, the arithmetic writhe and algebraic isotopies of space curves
• Quadratic counts of highly tangent lines to hypersurfaces
• ${\mathbb{A}}^1$-Brouwer degrees in Macaulay2