The concepts of tropical semiring and tropical hypersurface, are extended to the case of an arbitrary ordered group. Then, we define the tropicalization of a polynomial with coefficients in a Krull-valued field. After a close study of the properties of the operator “tropicalization" we conclude with an extension of Kapranov’s theorem to algebraically closed fields together with a valuation over an ordered group.
Les concepts de « semi-anneau » et d’« hypersurface tropicale » sont étendus au cas des groupes ordonnés quelconques. Ensuite, nous definissons la « tropicalisation » d’un polynôme à coefficients dans un corps valué. Après une étude détaillée de l’opérateur de tropicalisation, nous donnons une généralisation du théorème de Kapranov aux corps algébriquement clos munis d’une valuation à valeurs dans un groupe ordonné.
@article{AFST_2010_6_19_3-4_525_0,
author = {Fuensanta Aroca},
title = {Krull-Tropical {Hypersurfaces}},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {525--538},
publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
address = {Toulouse},
volume = {Ser. 6, 19},
number = {3-4},
year = {2010},
doi = {10.5802/afst.1255},
mrnumber = {2790807},
zbl = {1223.14069},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1255/}
}
TY - JOUR AU - Fuensanta Aroca TI - Krull-Tropical Hypersurfaces JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2010 SP - 525 EP - 538 VL - 19 IS - 3-4 PB - Université Paul Sabatier, Institut de Mathématiques PP - Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1255/ DO - 10.5802/afst.1255 LA - en ID - AFST_2010_6_19_3-4_525_0 ER -
%0 Journal Article %A Fuensanta Aroca %T Krull-Tropical Hypersurfaces %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2010 %P 525-538 %V 19 %N 3-4 %I Université Paul Sabatier, Institut de Mathématiques %C Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1255/ %R 10.5802/afst.1255 %G en %F AFST_2010_6_19_3-4_525_0
Fuensanta Aroca. Krull-Tropical Hypersurfaces. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 19 (2010) no. 3-4, pp. 525-538. doi : 10.5802/afst.1255. https://afst.centre-mersenne.org/articles/10.5802/afst.1255/
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