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Generalisations of Tropical Geometry over Hyperfields / JAMES MAXWELL

Swansea University Author: JAMES MAXWELL

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DOI (Published version): 10.23889/SUthesis.61752

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 descri...

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Published: Swansea 2022
Institution: Swansea University
Degree level: Doctoral
Degree name: Ph.D
Supervisor: Giansiracusa, Jeffrey ; Beggs, Edwin
URI: https://cronfa.swan.ac.uk/Record/cronfa61752
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first_indexed 2022-11-01T13:56:35Z
last_indexed 2023-01-13T19:22:43Z
id cronfa61752
recordtype RisThesis
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spelling v2 61752 2022-11-01 Generalisations of Tropical Geometry over Hyperfields 8a73ca359f44eea0390c8d73b9eb0821 JAMES MAXWELL JAMES MAXWELL true false 2022-11-01 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. E-Thesis Swansea Tropical, Hyperfield, Convexity, Variety, Polynomial, Algebra, Geometry 26 10 2022 2022-10-26 10.23889/SUthesis.61752 COLLEGE NANME COLLEGE CODE Swansea University Giansiracusa, Jeffrey ; Beggs, Edwin Doctoral Ph.D EPSRC DTP (EP/R51312X/1) EPSRC DTP (EP/R51312X/1) 2023-10-10T16:10:21.2060651 2022-11-01T13:49:41.2071513 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics JAMES MAXWELL 1 61752__25630__d14c69acafae419caaf7a350a214e41e.pdf Maxwell_James_PhD_Thesis_Final_Redacted_Signature.pdf 2022-11-01T14:38:18.9396667 Output 3570352 application/pdf E-Thesis – open access true Copyright: The author, James W. Maxwell, 2022. true eng
title Generalisations of Tropical Geometry over Hyperfields
spellingShingle Generalisations of Tropical Geometry over Hyperfields
JAMES MAXWELL
title_short Generalisations of Tropical Geometry over Hyperfields
title_full Generalisations of Tropical Geometry over Hyperfields
title_fullStr Generalisations of Tropical Geometry over Hyperfields
title_full_unstemmed Generalisations of Tropical Geometry over Hyperfields
title_sort Generalisations of Tropical Geometry over Hyperfields
author_id_str_mv 8a73ca359f44eea0390c8d73b9eb0821
author_id_fullname_str_mv 8a73ca359f44eea0390c8d73b9eb0821_***_JAMES MAXWELL
author JAMES MAXWELL
author2 JAMES MAXWELL
format E-Thesis
publishDate 2022
institution Swansea University
doi_str_mv 10.23889/SUthesis.61752
college_str Faculty of Science and Engineering
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hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
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description 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.
published_date 2022-10-26T16:10:23Z
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