### Generalising Kapranov’s Theorem for hyperfields

Algebra Seminar

14th December 2022, 2:30 pm – 3:30 pm

Fry Building, 2.04

We will begin by introducing Kapranov's theorem which is a foundational result in tropical geometry. It states that the set of tropicalisations of points on a hypersurface coincides precisely with the tropical variety of the tropicalisation of the defining polynomial. We then discuss the algebraic setting of hyperfields, which allow addition to be multivalued operation. We extend classical notions for polynomials over hyperfields, along with examples in this more general setting. We then present a specific class of hyperfields homomorphisms that satisfy a generalised version of Kapranov’s theorem.

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