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Positive tropicalizations of determinantal varieties and the Gondran–Minoux rank
Algebraic Combinatorics, Volume 8 (2025) no. 5, pp. 1349-1352

We point out to a connection of an old approach of Gondran and Minoux to ranks of matrices over the algebra $(\mathbb{R},\max ,+)$ and a recent work of Brandenburg, Loho, and Sinn on positive tropicalizations of algebraic varieties. This leads to examples of tropical varieties and their tropical bases which are not the corresponding positive tropical generating sets. In particular, we show that the $d\times d$ minors of a $d\times n$ matrix of variables form a positive tropical generating set if and only if either (1) $d=n$ or (2) $d\leqslant 4$.

Received:
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Accepted:
Published online:
DOI: 10.5802/alco.440
Classification: 14P05, 15A03
Keywords: Gondran–Minoux rank, tropical varieties
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Shitov, Yaroslav. Positive tropicalizations of determinantal varieties and the Gondran–Minoux rank. Algebraic Combinatorics, Volume 8 (2025) no. 5, pp. 1349-1352. doi: 10.5802/alco.440

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