Tropical toric geometry

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Functorial tropicalization of logarithmic schemes: the case of constant coefficients (2013, background): “Section 7, finally, is concerned with a comparison of our construction with the well-known embedded tropicalization for (subvarieties of) toric varieties in the sense of Kajiwara and Payne (see [29] and [41]) and a proof of Corollary 1.”
The tropical momentum map: a classification of toric log symplectic manifolds (2014, methodology): “2.19 The notion of tropical domain defined above is closely related to the notion of extended tropicalization Trop(X ) of a complex toric variety X introduced by Kajiwara [10] and Payne [17] (see also [5] for similar ideas).”