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Geometry of tropical extensions of hyperfields

Research output: Contribution to Journal/MagazineJournal articlepeer-review

E-pub ahead of print
<mark>Journal publication date</mark>26/11/2024
<mark>Journal</mark>Proceedings of the Royal Society of Edinburgh: Section A Mathematics
Number of pages42
Publication StatusE-pub ahead of print
Early online date26/11/24
<mark>Original language</mark>English

Abstract

We study the geometry of tropical extensions of hyperfields, including the ordinary, signed, and complex tropical hyperfields. We introduce the framework of ‘enriched valuations’ as hyperfield homomorphisms to tropical extensions and show that a notable family of them are relatively algebraically closed. Our main results are hyperfield analogues of Kapranov’s theorem and the Fundamental theorem of tropical geometry. Utilizing these theorems, we introduce fine tropical varieties and prove a structure theorem for them in terms of their initial ideals.