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Rational function semifields of tropical curves are finitely generated over the tropical semifield. (English) Zbl 1504.14106

Summary: We prove that the rational function semifield of a tropical curve is finitely generated as a semifield over the tropical semifield \(\boldsymbol{T}:=(\boldsymbol{R}\cup\{-\infty\},\max,+)\) by giving a specific finite generating set. Also, we show that for a finite harmonic morphism between tropical curves \(\varphi:\Gamma\to\Gamma'\), the rational function semifield of \(\Gamma\) is finitely generated as a \(\varphi^\ast(\mathrm{Rat}(\Gamma'))\)-algebra, where \(\varphi^\ast(\mathrm{Rat}(\Gamma'))\) stands for the pull-back of the rational function semifield of \(\Gamma'\) by \(\varphi\).

MSC:

14T10 Foundations of tropical geometry and relations with algebra
14T20 Geometric aspects of tropical varieties
15A80 Max-plus and related algebras
12K10 Semifields
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