Tropical Positivity and Determinantal Varieties

2022/05/30 by Brandenburg, Marie-Charlotte, Loho, Georg, Sinn, Rainer
#05E14 (Primary) 52B40 (Secondary) #14M12 #14P05 #14T15 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2205.14972

Abstract

We initiate the study of positive-tropical generators as positive analogues of the concept of tropical bases. Applying this to the tropicalization of determinantal varieties, we develop criteria for characterizing their positive part. We focus on the study of low-rank matrices, in particular matrices of rank 2 and 3. Moreover, in the case square-matrices of corank 1, we fully classify the signed tropicalization of the determinantal variety, even beyond the positive part.

Related

• Real and Positive Tropicalizations of Symmetric Determinantal Varieties
• Computing positive tropical varieties and lower bounds on the number of positive roots
• The Tropical Variety of Symmetric Rank 2 Matrices
• Signed tropicalization of polar cones
• Three notions of tropical rank for symmetric matrices