Triangulations of a product of two simplices and, more generally, of root polytopes are closely related to Gelfand-Kapranov-Zelevinsky's theory of discriminants, to tropical geometry, tropical oriented matroids, and to generalized permutohedra. We introduce a new approach to these objects, identifying a triangulation of a root polytope with a certain bijection between lattice points of two generalized permutohedra. In order to study such bijections, we define trianguloids as edge-colored graphs satisfying simple local axioms. We prove that trianguloids are in bijection with triangulations of root polytopes.

中文翻译：

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根多面体的三角面和三角剖分

两个单纯形乘积的三角剖分，更一般地说，根多面体的三角剖分与 Gelfand-Kapranov-Zelevinsky 的判别式理论、热带几何、热带定向拟阵以及广义置换面体密切相关。我们为这些对象引入了一种新方法，识别根多面体的三角剖分，并在两个广义置换面体的格点之间具有一定的双射。为了研究这种双射，我们将三角面定义为满足简单局部公理的边有色图。我们证明三角体与根多面体的三角剖分是双射的。