Ehrhart functions and symplectic embeddings of ellipsoids
Ehrhart functions and symplectic embeddings of ellipsoids
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DOI：
10.1112/jlms.12299
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发表时间：
2013-07
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影响因子：
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通讯作者：
Daniel Cristofaro-Gardiner;A. Kleinman
中科院分区：
文献类型：
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作者：
Daniel Cristofaro-Gardiner;A. Kleinman
McDuff先前已经证明，当且仅当某些组合准则成立时，一个四维辛椭球体可以辛嵌入另一个四维辛椭球体。我们用Ehrhart拟多项式理论重新解释了这个组合准则，并利用它给出了关于辛嵌入问题中“无限阶梯”存在性的McDuff-Schlenk定理和frenkel - m<s:1> ller定理的纯组合证明。然后我们发现了第三个新的阶梯，并推测这是嵌入有理椭球体的仅有的三个阶梯。还讨论了其他几个应用；例如，我们给出了Ehrhart函数呈现周期坍缩的三角形的新例子。
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.