Stephen McKean: Multivariate Bezoutians and A1-degrees

The theme of this talk is “the unreasonable effectiveness of commutative algebra in motivic homotopy theory.” While Morel’s GW(k)-valued Brouwer degree is an extremely useful tool, one would like to compute the degree in a formulaic way. Cazanave gave a formula for the global degree of endomorphisms of P^1 in terms of the Bézoutian, while Kass and Wickelgren gave a commutative algebraic formula for the local degree at rational points. After discussing these results, I will describe forthcoming work (joint with Thomas Brazelton and Sabrina Pauli) on computing the local and global degrees in terms of multivariate Bézoutians. Our results generalize Cazanave’s theorem on global degrees to higher dimensions, as well as remove any residue field assumptions on local degree computations.