We derive necessary and sufficient conditions for periodic and for elliptic periodic trajectories of billiards within an ellipse in the Minkowski plane in terms of an underlining elliptic curve. We provide several examples of periodic and elliptic periodic trajectories with small periods. We observe a relationship between Cayley-type conditions and discriminantly separable and factorizable polynomials. Equivalent conditions for periodicity and elliptic periodicity are derived in terms of polynomial-functional equations as well. The corresponding polynomials are related to the classical extremal polynomials. In particular, the light-like periodic trajectories are related to the classical Chebyshev polynomials. Similarities and differences with respect to the previously studied Euclidean case are highlighted.

中文翻译：

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Minkowski平面和Akhiezer多项式的圆锥曲线内的周期性台球

我们根据下划线的椭圆曲线，得出Minkowski平面中椭圆内台球的周期性和椭圆周期性轨迹的必要和充分条件。我们提供了具有小周期的周期和椭圆周期轨迹的几个示例。我们观察到Cayley型条件与判别式可分离和可分解多项式之间的关系。周期和椭圆周期的等效条件也根据多项式函数方程推导。相应的多项式与经典极值多项式有关。特别地，类似光的周期性轨迹与经典的切比雪夫多项式有关。强调了与先前研究的欧几里得案例的异同。