球面和双曲平面上台球的代数Birkhoff猜想
Algebraic Birkhoff conjecture for billiards on Sphere and Hyperbolic plane
摘要：
本文考虑球面或双曲平面上的凸曲线$γ$。研究了在$γ$有界区域内Birkhoff台球速度积分多项式的存在性问题.我们将结果推广了S. Bolotin（1992）的结果，并利用对偶曲线$Γ$得到了多项式可积性的新障碍。我们遵循S. Tabachnikov外台球在飞机上，并适用于后来在我们最近的文件，以伯克霍夫台球的帮助下，一个新的所谓的角度台球。
英文摘要：
We consider a convex curve $γ$ lying on the Sphere or Hyperbolic plane. We study the problem of existence of polynomial in velocities integrals for Birkhoff billiard inside the domain bounded by $γ$. We extend the result by S. Bolotin (1992) and get new obstructions on polynomial integrability in terms of the dual curve $Γ$. We follow a method which was introduced by S. Tabachnikov for Outer billiards in the plane and was applied later on in our recent paper to Birkhoff billiards with the help of a new the so called Angular billiard.
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