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Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Geometry of tropical extensions of hyperfields
AU - Maxwell, James
AU - Smith, Ben
PY - 2024/11/26
Y1 - 2024/11/26
N2 - 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.
AB - 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.
U2 - 10.48550/arXiv.2309.17302
DO - 10.48550/arXiv.2309.17302
M3 - Journal article
JO - Proceedings of the Royal Society of Edinburgh: Section A Mathematics
JF - Proceedings of the Royal Society of Edinburgh: Section A Mathematics
SN - 0308-2105
ER -