Studies on the geometry of discrete integrable systems and solvable chaotic systems

利用周期离散户田格的谱曲线与其它两条谱曲线的交点，建立了任意系统尺寸的周期离散户田格的几何实现。这种实现是由谱曲线的对称积上的点的加法来实现的，并利用周期离散户田格的守恒量具体给出了实现中的曲线，我们还通过热带平面曲线建立了周期离散户田格的超离散模拟，即任意系统尺寸的超离散周期户田格的几何实现.利用超离散周期户田格的守恒量，给出了实现中的热带曲线，其中之一是超离散周期户田格的谱曲线（热带超椭圆曲线）.

We establish a geometric realization of the periodic discrete Toda lattice of an arbitrary system size by using the intersection of its spectral curve and other two curves. This realization is reduced from the addition of points on the symmetric product of the spectral curve, and the curves in the realization are concretely given by using the conserved quantities of the periodic discrete Toda lattice.We also establish a geometric realization of the ultradiscrete periodic Toda lattice of an arbitrary system size, which is an ultradiscrete analogue of the periodic discrete Toda lattice, via tropical plane curves. The tropical curves in the realization, one of which is the spectral curve of the ultradiscrete periodic Toda lattice (a tropical hyperelliptic curve), are also given by using the conserved quantities of the ultradiscrete periodic Toda lattice.