McDuff has previously shown that one four‐dimensional symplectic ellipsoid can be symplectically embedded into another if and only if a certain combinatorial criteria holds. We reinterpret this combinatorial criteria using the theory of Ehrhart quasipolynomials, and we use this to give purely combinatorial proofs of theorems of McDuff–Schlenk and Frenkel–Müller, concerning the existence of ‘infinite staircases’ in symplectic embedding problems. We then find a third, new, staircase and conjecture that these are the only three staircases for embeddings into rational ellipsoids. Several other applications are also discussed; for example, we give new examples of triangles whose Ehrhart function exhibits a period collapse.

中文翻译：

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椭球的Ehrhart函数和辛嵌入

麦克杜夫（McDuff）先前已经证明，只要且仅当满足某种组合条件时，一个四维辛椭圆形体才能被嵌入到另一个椭圆形中。我们使用Ehrhart拟多项式理论重新解释了这种组合准则，并以此纯粹给出了McDuff-Schlenk和Frenkel-Müller定理的组合证明，其中涉及辛嵌入问题中“无限阶梯”的存在。然后，我们发现第三个，新的楼梯和猜想，它们是将嵌入椭圆形的唯一三个楼梯。还讨论了其他几个应用程序；例如，我们给出了Ehrhart函数表现出周期崩溃的三角形的新示例。