We study real and positive tropicalizations of the varieties of low rank symmetric matrices over real or complex Puiseux series. We show that real tropicalization coincides with complex tropicalization for rank two and corank one cases. We also show that the two notions of positive tropicalization introduced by Speyer and Williams coincide for symmetric rank two matrices, but they differ for symmetric corank one matrices.
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Keywords: ranks of tropical matrices, positive tropicalization, symmetric matrices, determinantal varieties, bicolored trees
Al Ahmadieh, Abeer 1 ; Cai, May 2 ; Yu, Josephine 2
CC-BY 4.0
@article{ALCO_2026__9_1_1_0,
author = {Al Ahmadieh, Abeer and Cai, May and Yu, Josephine},
title = {Real and {Positive} {Tropicalizations} of {Symmetric} {Determinantal} {Varieties}},
journal = {Algebraic Combinatorics},
pages = {1--19},
year = {2026},
publisher = {The Combinatorics Consortium},
volume = {9},
number = {1},
doi = {10.5802/alco.462},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.462/}
}
TY - JOUR AU - Al Ahmadieh, Abeer AU - Cai, May AU - Yu, Josephine TI - Real and Positive Tropicalizations of Symmetric Determinantal Varieties JO - Algebraic Combinatorics PY - 2026 SP - 1 EP - 19 VL - 9 IS - 1 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.462/ DO - 10.5802/alco.462 LA - en ID - ALCO_2026__9_1_1_0 ER -
%0 Journal Article %A Al Ahmadieh, Abeer %A Cai, May %A Yu, Josephine %T Real and Positive Tropicalizations of Symmetric Determinantal Varieties %J Algebraic Combinatorics %D 2026 %P 1-19 %V 9 %N 1 %I The Combinatorics Consortium %U https://alco.centre-mersenne.org/articles/10.5802/alco.462/ %R 10.5802/alco.462 %G en %F ALCO_2026__9_1_1_0
Al Ahmadieh, Abeer; Cai, May; Yu, Josephine. Real and Positive Tropicalizations of Symmetric Determinantal Varieties. Algebraic Combinatorics, Volume 9 (2026) no. 1, pp. 1-19. doi: 10.5802/alco.462
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