Positive tropicalizations of determinantal varieties and the Gondran–Minoux rank

We point out to a connection of an old approach of Gondran and Minoux to ranks of matrices over the algebra <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ℝ</mml:mi> <mml:mo>,</mml:mo> <mml:mo form="prefix" movablelimits="true">max</mml:mo> <mml:mo>,</mml:mo> <mml:mo>+</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>×</mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:math> minors of a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>×</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> matrix of variables form a positive tropical generating set if and only if either (1) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> or (2) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>⩽</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> .