Abstract We show that the Chow covectors of a linkage matching field define a bijection between certain degree vectors and lattice points, and we demonstrate how one can recover the linkage matching field from this bijection. This resolves two open questions from Sturmfels and Zelevinsky (1993) [26] on linkage matching fields. For this, we give an explicit construction that associates a bipartite incidence graph of an ordered partition of a common set to each lattice point in a dilated simplex. Given a triangulation of a product of two simplices encoded by a set of spanning trees on a bipartite node set, we similarly prove that the bijection from left to right degree vectors of the trees is enough to recover the triangulation. As additional results, we show a cryptomorphic description of linkage matching fields and characterise the flip graph of a linkage matching field in terms of its prodsimplicial flag complex. Finally, we relate our findings to transversal matroids through the tropical Stiefel map.

中文翻译：

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单纯形的匹配域和格点

摘要 我们展示了链接匹配域的 Chow 协向量定义了某个度向量和格点之间的双射，并演示了如何从这个双射中恢复链接匹配场。这解决了 Sturmfels 和 Zelevinsky (1993) [26] 关于链接匹配字段的两个开放性问题。为此，我们给出了一个显式构造，它将公共集合的有序分区的二分关联图与扩张单纯形中的每个格点相关联。给定由二部节点集上的一组生成树编码的两个单纯形的乘积的三角剖分，我们同样证明了从树的左到右度向量的双射足以恢复三角剖分。作为额外的结果，我们展示了链接匹配字段的加密描述，并根据其 prodsimplicial flag complex 来表征链接匹配字段的翻转图。最后，我们通过热带 Stiefel 地图将我们的发现与横向拟阵联系起来。