Generalisations of Tropical Geometry over Hyperfields

Abstract

Hyperfields are structures that generalise the notion of a field by way of allowing the addition operation to be multivalued. The aim of this thesis is to examine generalisations of classical theory from algebraic geometry and its combinatorial shadow, tropical geometry. We present a thorough description of the hyperfield landscape, where the key concepts are introduced. Kapranov’s theorem is a cornerstone result from tropical geometry, relating the tropicalisation function and solutions sets of polynomials. We generalise Kapranov’s Theorem for a class of relatively algebraically closed hyperfield homomorphisms. Tropical ideals are reviewed and we propose the property of matroidal equivalence as a method of associating the geometric objects defined by tropical ideals. The definitions of conic and convex sets are appropriately adjusted allowing for convex geometry over ordered hyperfields to be studied.